\def\multi#1{\vbox{\baselineskip=0pt\halign{\hfil$\scriptstyle\vphantom{(_)}##$\hfil\cr#1\crcr}}} \catcode`@=11 \def\dsquad{\quad} \def\dsmatrix#1{\null\,\vcenter{\normalbaselines\m@th\baselineskip=1.5\baselineskip \ialign{\hfil$\displaystyle##$\hfil&&\dsquad\hfil$\displaystyle##$\hfil\crcr \mathstrut\crcr\noalign{\kern-\baselineskip} #1\crcr\mathstrut\crcr\noalign{\kern-\baselineskip}}}\,} \catcode`@=12 % at signs are no longer letters %%\overfullrule=0pt \baselineskip 15pt \settabs 8 \columns \+ \hfill&\hfill \hfill & \hfill \hfill & \hfill \hfill & \hfill \hfill & & \hfill \hfill\cr %\voffset=1truein tells how low in the page should we start! %\hoffset=.5truein %tells how far to right should we start! \parindent=.5truein %\hfuzz=3.44182pt %\hsize 6truein %\vsize=8truein %\hsize=14cm\vsize=23.5cm \hsize=13.8cm\vsize=21.5cm \hoffset=1.25cm % Twelve point fonts: \def\smagstep{scaled \magstep} \font\twlsc = cmcsc10 \smagstep 1 % small caps \font\twlrm = cmr10 \smagstep 1 % roman \font\twlit = cmti10 \smagstep 1 % text italic \font\twlsl = cmsl10 \smagstep 1 % slanted \font\twlbf = cmbx10 \smagstep 1 % extended bold \font\twlmi = cmmi10 \smagstep 1 % math italic \skewchar\twlmi ='177 % for placement of accents \font\twlsy = cmsy10 \smagstep 1 % math symbols \skewchar\twlsy ='60 % for placement of math accents \font\twltt = cmtt10 \smagstep 1 % typewriter \hyphenchar\twltt = -1 % suppress hyphenation in \tt font \font\tensc = cmcsc10 % small caps \font\ninbf = cmbx9 % extended bold \font\egtrm = cmr8 % roman \font\egtit = cmti8 % text italic \font\egtsl = cmsl8 % slanted \font\egtbf = cmbx8 % extended bold \font\egtmi = cmmi8 % math italic \skewchar\egtmi ='177 % for placement of accents \font\egtsy = cmsy8 % math symbols \skewchar\egtsy ='60 % for placement of math accents \font\sixrm = cmr6 % roman \font\sixit = cmti7 scaled 857 % text italic \font\sixsl = cmsl10 scaled 600 % slanted \font\sixbf = cmbx7 scaled 857 % extended bold \font\sixmi = cmmi6 % math italic \skewchar\sixmi ='177 % for placement of accents \font\sixsy = cmsy6 % math symbols \skewchar\sixsy ='60 % for placement of math accents \font\ninei=cmmi9 \skewchar\ninei='177 \font\sixi=cmmi6 \skewchar\sixsy='60 \font\ninesy=cmsy9 \skewchar\ninesy='60 \font\nineit=cmti9 \font\ninesl=cmsl9 \font\ninebf=cmbx9 \font\ninett=cmtt9 \def\ninepoint{\def\rm{\fam0\ninerm}% \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1=\fivei \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3=\tenex \def\it{\fam\itfam\nineit}% \textfont\itfam=\nineit \def\sl{\fam\slfam\ninesl}% \textfont\slfam=\ninesl \def\bf{\fam\bffam\ninebf}% \textfont\bffam=\ninebf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\tt{\fam\ttfam\ninett}% \textfont\ttfam=\ninett % \tt \ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt % \def\MF{{\manual hijk}\-{\manual lmnj}}% \let\sc=\sevenrm \let\big=\ninebig \setbox\strutbox=\hbox{\vrule height8pt depth3pt width0pt}% \normalbaselines\rm} \catcode`@=11 \def\ninebig#1{{\hbox{$\textfont0=\tenrm\textfont2=\tensy \left#1\vbox to7.25pt{}\right.\n@space$}}} \catcode`@=12 \def\twelvepoint{ \def\rm{\fam0\twlrm} \textfont0=\twlrm \scriptfont0\egtrm \scriptscriptfont0=\sixrm \textfont1=\twlmi \scriptfont1\egtmi \scriptscriptfont1=\sixmi \textfont2=\twlsy \scriptfont2\egtsy \scriptscriptfont2=\sixsy \textfont3=\tenex \scriptfont3\tenex \scriptscriptfont3=\tenex \textfont\itfam=\twlit \def\it{\fam\itfam\twlit} \textfont\slfam=\twlsl \def\sl{\fam\slfam\twlsl} \textfont\ttfam=\twltt \def\tt{\fam\ttfam\twltt} \textfont\bffam=\twlbf \scriptfont\bffam=\ninbf \scriptscriptfont\bffam=\ninbf \def\bf{\fam\bffam\twlbf} \normalbaselineskip=14.4pt \setbox\strutbox=\hbox{\vrule height.7\baselineskip depth.3\baselineskip width0pt} } \twelvepoint\rm % Adriano 10 pt: %\font\title=cmbx10 scaled\magstep2 %\font\normal=cmr10 %\font\small=cmr6 \font\ninerm=cmr9 %\font\ita=cmti10 %\font\bol=cmbx10 \def\normal{\rm} %\let\ninerm=\relax \def\ita{\it} \def\bol{\bf} \def\del {\partial} \def \con {\subseteq} \def \scong {\ess \cong \ess } \def \brkpg {\supereject} \def\lll{\lambda} \def\La {\Lambda} \def\sig{\sigma} \def\dia{\diamondsuit} \def \ggg {\gamma} \def \GG {\Gamma} \def\PI{\Pi} %\def \-> {\rightarrow} \def\LL{\big\langle} \def \ra {\rightarrow} \def\RR {\big\rangle} \def\splangle{\langle\ssp } \def\sprangle {\ssp \rangle} \def \UU {{\scriptscriptstyle U} } \def \HHH {{\bf H}} \def\OO {\Omega} \def\DD {\Delta} \def\OM {\Omega} \def\TOM {\tilde {\Omega}} \def\om {\omega} \def\la {\lambda} \def \RA {\rightarrow} \def \LA {\lefttarrow} \def\xon {x_1,x_2,\ldots ,x_n} \def\GA {\Gamma} \def \sas {\vskip .06truein} \def\sa{{\vskip .125truein}} \def\msas{{\vskip -.01in}} \def\sap{{\vskip .25truein}} \def\sapp {{\vskip .5truein}} \def\sss {\sigma} \def \eee {\epsilon} \def\DDD {\Delta} \def\aaa {\alpha} \def\bbb {\beta} \def\ddd{\delta} \def\ttt{\theta} \def\ggg {\gamma} \def\aa {\alpha} \def\bb {\beta} \def\dd{\delta} \def\gg {\gamma} \def\con {\subseteq} \def \ses {\enskip = \enskip} \def \sps {\enskip + \enskip} \def \ssms {\, -\,\,} \def \sses {\, =\,\,} \def \sscs {\, ,} \def \ssps {\,+\, \,} \def \sms {\enskip -\enskip} \def \lag {\langle\ssp } \def \shhh {{\scriptstyle \cup\hskip -.18em\cup}} \def \rag {\ssp \rangle} \def \scs {\ssp , \ssp} \def \ess {\enskip} \def \ssp {\hskip .25em} \def \bigsp {\hskip .5truein} \def \part {\vdash} \def \scos {\ssp :\ssp} \def \DD {\Delta} \def \X {{X}} \def \Y {{Y}} \def \Sl {S_\lambda} \def \oom {{\omega}} \normal \def\uul{\underline} %\normalbaselineskip=12pt 18pt for space and a half, %\normalbaselines24pt for double space \sap \def\today{\ifcase\month\or January\or February\or March\or April\or may\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day, \number\year} \def \LRA {{\ssp \longrightarrow \ssp}} \def \RA {{ \rightarrow }} \def \LA {{ \leftarrow }} \def \BH {{\bf H}} \def \CC {{\cal C}} \def \CR {{\cal R}} \def \BV {{\bf V}} \def \BR {{\bf R}} \def \BQ {{\bf Q}} \def \BM {{\bf M}} \def \BPI {{\bf \PI'}} \def \DA {{\downarrow}} \def \TM {{\tilde M}} \def \om {\omega} \def \BH {{\bf H}} \def \TK {{\tilde K}} \def \Om {\Omega} \def \bk {{\bf k}} \def \TH {{\tilde H}} \def \th {{\tilde h}} \def \xon {x_1,\ldots ,x_n} \def \yon {y_1, \ldots ,y_n} \def \LLL {\langle} \def \RRR {\rangle} \def \BH {{\bf H}} \def \BR {{\bf R}} \def \BQ {{\bf Q}} \def \TAU {{\cal T}} \def \PAU {{\cal P}} \def \uS {{\underline S}} \def \ue {{\underline e}} \def \LA {\Lambda} \def \TS {\textstyle} \def \om {\omega} \def \TK {{\tilde K}} \def \TH {{\tilde H}} \def \xon {x_1,\ldots ,x_n} \def \yon {y_1, \ldots ,y_n} \def \LLL {\langle} \def \RRR {\rangle} \def \xom {x_1,\ldots ,x_m} %\def \sgmu {\ ^{sg}\BH_\mu} \def \bhmu {\BH_\mu} \def \pmu {\rho_\mu} \def \FP {{\cal FP}[X,Y]} \def \gamu {{\bf \Gamma}_\mu} \def \mgamu {m({\bf \Gamma}_\mu)} \def \RImu {{\cal RI}_\mu} \def \da {\downarrow} \def \sda {\hskip -.03truein\downarrow} \def \dan {\hskip -.03truein\downarrow_n} \def \ua {\uparrow} \def \sua {\hskip -.03truein\uparrow} \def \os {\overline {s}} \def \ox {\overline {x}} \def \uan {\hskip -.03truein\uparrow^n} %\def\today{\ifcase\month\or %January\or February\or March\or April\or May\or June\or %July\or August\or September\or October\or November\or %December\fi %\space\number\day, \number\year} %\headline={\small Explicit formulas for q,t -Kotska $\ess\ess\ess\ess\ess$ \hfill$\ess\ess\ess$\today$\ess\ess\ess\ess\ess\ess$ \folio } \footline={\hfil} \headline={\tensc \ifodd\pageno \ifnum\pageno>1 \hfill EXPLICIT FORMULAS FOR $q,t$-KOSTKA COEFFICIENTS \hfill \else \hfill \fi \else \hfill GARSIA, HAIMAN, AND TESLER \hfill \fi } \footline={\hss\twlrm\folio\hss} \def\FOOTNOTE#1#2{{\ninepoint\parindent=1em\footnote{#1}{#2}}} %\let\FOOTNOTE\footnote % headings without page break -- kludge \def\heading#1{\noindent{#1}\par\nobreak} %\sap \centerline {\bol Explicit Plethystic Formulas } \centerline {\bol for } \centerline {\bol Macdonald q,t-Kostka Coefficients } \sa \centerline {\ita A.\ M.\ Garsia${}^{\dag}$\FOOTNOTE{} {$\dag$\ess Work carried out under NSF grant support.}, M.\ Haiman${}^{\dag}$, and G.\ Tesler${}^{\dag}$} \sa {\par\ninepoint\rm \narrower \noindent{\bol Abstract. } For a partition $\mu=(\mu_1>\mu_2>\cdots >\mu_k>0)$ set $ B_\mu(q,t)=\sum_{i=1}^k\, t^{ i-1 }\,(1+ \cdots + q^{\mu_i-1}$). In [8] Garsia-Tesler proved that if $\gamma$ is a partition of $k$ and $\la=(n-k,\gamma)$ is a partition of $n$, then there is a unique symmetric polynomial $k_\gamma(x;q,t)$ of degree $\leq k$ with the property that $\TK_{\la\mu}(q,t)=k_\gamma[B_\mu(q,t);q,t]$ holds true for all partitions $\mu$. It was shown there that these polynomials have Schur function expansions of the form $k_\gamma(x;q,t)= \sum_{|\rho|\leq |\gamma|}S_\la(x)\, \, k_{\rho,\gamma}(q,t)$ where the $ k_{\rho,\gamma}(q,t)$ are polynomials in $q,t,1/q,1/t$ with integer coefficients. This result yielded the first proof of the Macdonald polynomiality conjecture. It also was used in a proof [7] of the positivity conjecture for the $\TK_{\la\mu}(q,t)$ for any $\la$ of the form $\la=(r,2,1^m)$ and arbitrary $\mu$. In this paper we show that the polynomials $k_\gamma(x;q,t)$ may be given a very simple explicit expression in terms of the operator $\nabla$ studied in [2]. In particular we also obtain a new proof of the polynomiality of the coefficients $\TK_{\la\mu}(q,t)$. Further byproducts of these developments are a new explicit formula for the polynomial $\TH_\mu[X;q,t]=\sum_\la\, S_\la[X]\TK_{\la\mu}(q,t)$ and a new derivation of the symmetric function results of Sahi [16] and Knop [11], [12]. }% end narrower \sa \heading{\bol Introduction} To state our results we need to review some notation and recall some basic facts. We work with the algebra $\La$ of symmetric functions in a formal infinite alphabet $X=x_1,x_2,\ldots,$ with coefficients in the field of rational functions ${\bf Q}(q,t)$. We also denote by $\La_{Z[q,t\, ]}$ the algebra of symmetric functions in $X $ with coefficients in $Z[q,t\, ]$. We write $\La^{=d}$ for the space of symmetric functions homogeneous of degree $d$. The spaces $\La^{\leq d}$ and $\La^{>d}$ are analogously defined. We shall make extensive use here of ``{\ita plethystic\/}'' notation. This is a notational device which simplifies manipulation of symmetric function identities. It can be easly defined and programmed in $\scriptstyle MATHEMATICA$ or $\scriptstyle MAPLE$ if we view symmetric functions as formal power series in the power symmetric functions $p_k$. To begin with, if $E=E[t_1,t_2,t_3,\ldots ]$ is a formal Laurent series in the variables $t_1,t_2,t_3,\ldots $ (which may include the parameters $q,t$) we set $$ p_k[E]\ses E[t_1^k,t_2^k,t_3^k,\ldots ]\ess . $$ More generally, if a certain symmetric function $F$ is expressed as the formal power series $$ F\ses Q[p_1,p_2,p_3,\ldots ] $$ then we simply let $$ F[E]\ses Q[p_1,p_2,p_3,\ldots ]\Big|_{p_k \RA E[t_1^k,t_2^k,t_3^k,\ldots ]}\ess . \eqno {\rm I}.1 $$ and refer to it as ``{\ita plethystic substitution\/}'' of $E$ into the symmetric function $F$. We make the convention that inside the plethystic brackets ``$[\,\,]$'', $X$ and $X_n$ respectively stand for $x_1+x_2+x_3+\cdots$ and $x_1+x_2 +\cdots +x_n$. In particular, one sees immediately from this definition that if $f(x_1,x_2 ,\ldots,x_n)$ is a symmetric function then $f[X_n]=f(x_1,x_2 ,\ldots,x_n)$. We shall also make use of the symbol $\OM(x)$ to represent the symmetric function $$ \OM(x)\ses \prod_{i\geq 1} {1\over 1-x_i }\ess . $$ It is easily seen that in terms of it the Cauchy, Hall-Littlewood and Macdonald kernels may be respectively be given the compact forms $$ \OM[X_nY_m]\ess \scs\ess \ess \OM[X_nY_m(1-t)] \ess\ess\ess \hbox{and}\ess\ess\ess \OM[X_nY_m{\textstyle{ 1-t\over 1-q}}]\ess. $$ Indeed, since we may write $$ \OM \ses \exp\Big(\sum_{k\geq 1} {p_k \over k}\Big)\ess , $$ we see that the definition in I.2 gives $$ \OM[X_nY_m]=\prod_{i=1}^n\prod_{j=1} ^m {1\over 1-x_iy_j} \ess\ess\scs\ess\ess\ess \OM[X_nY_m(1-t)]=\prod_{i=1}^n\prod_{j=1} ^m {1-t \, x_iy_j\over 1-x_iy_j} $$ and $$ \OM[X_nY_m{\textstyle{ 1-t\over 1-q}}]\ses \prod_{i=1}^n\prod_{j=1} ^m \prod_{k=0}^\infty {1-t \,q^k x_iy_j\over 1-q^kx_iy_j} \ess . $$ In using plethystic notation we are forced to distinguish between two different minus signs. Indeed note that the definition in I.1 yields that we have $$ p_k[-X_n] \ses p_k[-x_1-x_2-\cdots -x_n]\ses -x_1^k-x_2^k-\cdots -x_n^k\ses -p_k[X_n]\ess . $$ On the other hand, on using the ordinary meaning of the minus sign, we would obtain $$ p_k[ X_n]\ssp \big|_{x_i\RA -x_i}\ses (-1)^k\ssp p_k[X_n]\ess . $$ Since both operations will necessarily occur in our formulas, we shall adopt the convention that when a certain variable has to be replaced by its negative, in the ordinary sense, then that variable will be prepended by a superscripted minus sign. For example, note that the $\om$ involution, which is customarily defined as the map which interchanges the elementary and homogeneous bases, may also be defined by setting $$ \om\, p_k\ses (-1)^{k-1}\, p_k\ess . $$ However, note that by the above conventions we obtain that $$ p_k[-^-X_n]\ses (-1)^{k-1}\, p_k[X_n]\ess . $$ In particular, for any symmetric polynomial $P$ of degree $\leq n$, we may write $$ \om\, P[X_n]\ses P[-^-X_n]\ess . \eqno {\rm I}.2 $$ Sometimes it will be convenient to use the symbol ``$\eee$'' to represent ${}^-1$. The idea is that we should treat $\ssp \eee\ssp $ as any of the other variables in carrying out plethystic operations and only at the end replace $\ssp \eee\ssp $ by $-1$ in the ordinary sense. A partition $\mu$ will be represented and identified with its Ferrers diagram. We shall use the French convention here and, given that the parts of $\mu$ are $\mu_1\geq \mu_2\geq \cdots\geq \mu_k>0$, we let the corresponding Ferrers diagram have $\mu_i$ lattice cells in the $i^{th}$ row (counting from the bottom up). It will be convenient to let $|\mu|$ and $l(\mu)$ denote respectively the sum of the parts and the number of nonzero parts of $\mu$. In this case $|\mu|=\mu_1+\mu_2+\cdots +\mu_k$ and $l(\mu)=k$. As customary the symbol ``$\mu \part n$'' will be used to indicate that $|\mu|=n\ssp $. Following Macdonald, the {\ita arm, leg, coarm\/} and {\ita coleg\/} of a lattice square $s$ are the parameters $a_\mu(s),l_\mu(s),a_\mu'(s)$ and $l_\mu'(s)$ giving the number of cells of $\mu$ that are respectively {\ita strictly} {\ninerm EAST, NORTH, WEST} and {\ninerm SOUTH} of $s$ in $\mu$. This given, here and after, for a partition $\mu=(\mu_1,\mu_2,\ldots ,\mu_k)$ we set $$ n(\mu)=\sum_{i=1}^k (i-1)\mu_i\ses \sum_{s\in \mu}\, l'_\mu(s)\ses \sum_{s\in \mu}\, l_\mu(s)\ess . $$ If $s$ is a cell of $\mu$ we shall refer to the monomial $w(s)=q^{a_\mu'(s)} t^{l_\mu' (s)} $ as the {\ita weight} of $s$. The sum of the weights of the cells of $\mu$ will be denoted by $B_\mu(q,t)$ and will be called the {\ita biexponent generator of }$\mu$. Note that we have \def \II {{\rm I}} $$ B_\mu(q,t)\ses \sum_{s\in \mu}\ssp q^{a_\mu'(s)} t^{l_\mu' (s)}\ses \sum_{i\geq 1} t^{i-1}\ssp {1-q ^{\mu_i}\over 1-q\ess\ess}\ess . \eqno {\rm I}.3 $$ If $\gamma\part k$ and $n-k \geq \max( \gamma) $, the partition of $n$ obtained by prepending a part $n-k$ to $\gamma$ will be denoted by $(n-k,\gamma )$. It will also be convenient to set $$ T_\mu\,=\, t^{n(\mu)}q^{n(\mu')}\,=\, \prod_{s\in\mu} q^{a_\mu'(s)} t^{l_\mu' (s)} \ess\ess \hbox{and}\ess D_\mu \,=\, (1-t)(1-q)B_\mu(q,t)-1\,.\;\; \eqno \II.4 $$ \sas We shall work here with the symmetric polynomial $\TH_\mu[X;q,t]$ with Schur function expansion $$ \TH_\mu[X;q,t]\ses \sum_{\la}\, S_\la[X]\, \TK_{\la\mu}(q,t)\ess , \eqno \II.5 $$ where the coefficients $\TK_{\la\mu}(q,t)$ are obtained from the Macdonald $q,t$-Kostka coefficients by setting $$ \TK_{\la\mu}(q,t)\ses t^{n(\mu)}\, K_{\la\mu}(q,1/t)\ess . \eqno \II.6 $$ As we shall see, most of the properties of $\TH_\mu[X;,q,t]$ we will need here can be routinely derived from the corresponding properties of the Macdonald's integral form $J_\mu[X;q,t]\, $ $(\dag)$ \FOOTNOTE{}{$(\dag)$\ess [15] Ch.\ VI, (8.3)}, via the identity $$ \TH_\mu[X;q,t]=t^{n(\mu)}J_\mu[{\textstyle{X\over 1-1/t}}\,; q,1/t\,]\ess . \eqno \II.7 $$ This polynomial occurs naturally in our previous work, where it is conjectured to give a representation theoretical interpretation to the coefficients $\TK_{\la\mu}(q,t)$. Another important ingredient in the present developments is the linear operator $\nabla$ defined, in term of the basis $\{\TH_\mu[X;q,t]\}_\mu\, $, by setting $$ \nabla\, \TH_\mu[X;q,t]\ses T_\mu\, \TH_\mu[X;q,t]\ess . \eqno \II.8 $$ This operator also plays a crucial role in the developments relating Macdonald polynomials to symmetric group representation theory [1], [3], [4], [5], [6] and to geometry [9]. Computer experimentation with $\nabla$ revealed that it has some truly remarkable properties. The reader is referred to [2] for a collection of results and conjectures about $\nabla$ that have emerged in the few years since its discovery. \sas It was shown in [8] that for any given $\gamma\part k$, there is a unique symmetric polynomial $\bk_\gamma(x;q,t)$ of degree $\ssp \leq k\ssp $ yielding $$ {\tilde K}_{(n-k,\gamma ),\mu}(q,t)\;=\; \bk_\gamma[\ssp B_\mu(q,t)\ssp;q,t ] \quad\quad (\ess \forall\ess\ess\mu\part n \geq k+ \max( \gamma)\ess ) \ess . \eqno \II.9 $$ Although a formula for $\bk_\gamma(x;q,t)$ could be extracted from the original proof of this results (see [8] Th. 4.1), it was of such complexity that it yielded very little information about the true nature of this polynomial. All that could be derived there is that $\bk_\gamma(x;q,t)$ has a Schur function expansion of the form $$ \bk_\gamma(x;q,t)\ses \sum_{|\rho|\leq k}\ssp S_\rho\ssp \bk_{\rho\gamma}(q,t) \eqno \II.10 $$ with each $\bk_{\rho\gamma}(q,t)$ a Laurent polynomial in $q,t$ with integer coefficients. This result was sufficient to prove the integral polynomiality of the $K_{\la\mu}(q,t)$. Moreover, a relatively small number of these polynomials already permitted the computation of extensive tables of the polynomials $\TH_\mu[X;q,t]$. \sa The remarkable development here is that, in terms of $\nabla$, the polynomial $\bk_\gamma(x;q,t)$ may be given a surprisingly simple expression. \sas %%%%\vbox{ \heading{\bol Theorem I.1} \sas {\ita For each $\gamma\part k$ let $$ \bk_\gamma '(x;q,t)\ses \nabla^{-1}\ssp S_\gamma\big[ {\textstyle{1-\ ^-X\over (1-t)(1-q)}}-1\big]\ess . \ess\ess\ess (\dag) \eqno \II.11 $$ \FOOTNOTE{}{$(\dag)\ess $ Here and in the following plethysms are to be carried out before operator actions.} Then $$ {\tilde K}_{(n-k,\gamma ),\mu}(q,t)\ses \bk_\gamma'[\ssp D_\mu(q,t)\ssp;q,t ] \quad\quad (\; \forall\ess\mu\part n \geq k+ \max( \gamma)\ess ) \; . \eqno \II.12 $$ In particular the symmetric polynomial uniquely characterized by I.9 and I.10 is given by the formula $$ \bk_\gamma[X]\ses \bk_\gamma'[(1-t)(1-q)X-1] \eqno \II.13 $$ } %%%%} Let us recall that the Hall scalar product for symmetric functions is defined by setting for the power basis $\{p_\rho\}_\rho$ $$ \LL p_{\rho^{(1)}}\scs p_{\rho^{(2)}} \RR\ses \cases{ z_\rho & if $\rho^{(1)}=\rho^{(2)}=\rho$ \cr\cr 0 & otherwise \cr } $$ where for a partition $\rho=1^{\aaa_1},2^{\aaa_2},3^{\aaa_3},\cdots$ we set as customary $$ z_\rho=1^{\aaa_1}2^{\aaa_2}3^{\aaa_3}\cdots \aaa_1!\aaa_2!\aaa_3!\cdots\ess . $$ We shall also need here the scalar product $\LL\scs \RR_*$ defined by setting $$ \LL p_{\rho^{(1)}}\scs p_{\rho^{(2)}} \RR_* \,=\, \cases{ (-1)^{ |\rho|-l(\rho) } \, z_\rho\, p_\rho[{(1-t)(1-q)}] & if $\rho^{(1)}=\rho^{(2)}=\rho\, $, \cr\cr 0 & otherwise. \cr } \hskip-2pt \eqno \II.14 $$ It will be convenient, here and in the following, to set for every $F[X ]\in \LA\,$, $$ F^*[X ]\ses F\big[{\textstyle{X\over(1-t)(1-q) }} ]\ess . $$ Our main object here is the following very general result which has a variety of important consequences including our formula I.11: \sas \heading{\bol Theorem I.2} \sas {\ita For each symmetric polynomial $f$ set $$ \BPI_f [X;q,t]\ses \nabla^{-1}\, f[X-^-1] \eqno \II.15 $$ Then for all $\mu$ we have $$ {\bf \PI}_f'[D_\mu;q,t]\ses \LL f\scs \TH_\mu[X+1]\RR_* \eqno \II.16 $$ Alternatively, if $f$ is homogeneous of degree $k$ and we also set $$ {\bf \PI}_f[X;q,t]\ses \nabla^{-1}\, f[{\textstyle{1-^-X\over (1-t)(1-q)}}]\ess , \eqno \II.17 $$ then for all $\mu\part n\geq k$ we have $$ \eqalign{ &a)\ess\ess\LL e_{n-k}^* \, f\sscs \TH_\mu\, \RR_* \ses {\bf \PI}_f' [D_\mu;q,t]\cr &b)\ess\ess\LL h_{n-k} \, f\sscs \TH_\mu\, \RR \,\ses {\bf \PI}_f [D_\mu;q,t]\ess .\cr } \eqno \II.18 $$ } \sas We can define a skew version $\TH_{\mu/\nu}$ of the symmetric polynomial $\TH_\mu$ yielding the addition formula $$ \TH_\mu[X+Y;q,t]\ses \sum_{\nu\con \mu}\, \TH_\nu[X ;q,t]\, \TH_{\mu/\nu}[Y;q,t]\ess . \eqno \II.19 $$ This can be derived from the analogous result for the Macdonald polynomial $Q_\la[X;q,t\, ]$ (see Ch.\ VI (7.9)). Now it develops that the identity in I.16 (with $\BPI_f$ given by I.15) is equivalent to the following truly remarkable formula yielding the polynomial $ \TH_\mu$. \sas %%%%\vbox{ \heading{\bol Theorem I.3} $$ \TH_\mu[X+1 \, ;q,t\, ]\ses \Om \big[{\textstyle{X \over M}}\big ]\, \nabla^{-1} \om \, \Om[{\textstyle{X \, D_\mu \over M}}\big] \eqno \II.20 $$ {\ita with} $$ M\ses (1-t)(1-q) \eqno \II.21 $$ Another corollary of Theorem I.2 may be stated as follows. \sas %%%%} %%%%\vbox{ \heading{\bol Theorem I.4} {\ita For a partition $\mu$ set $$ \delta_\mu[X;q,t\, ]\ses {\nabla^{-1}\TH_\mu[X-^-1] \over \th_\mu(q,t) \, \th_\mu'(q,t) } \eqno \II.22 $$ with $$ \th_\mu(q,t)=\prod_{s\in \mu}(q^{a_\mu(s)}-t^{l_\mu(s)+1}) \ess \scs \ess\ess \th_\mu'(q,t)=\prod_{s\in \mu}(t^{l_\mu (s)}-q^{a_\mu (s)+1})\,\, . \eqno \II.23 $$ Then $$ \delta_\mu[D_\la\, ;q,t\, ]\ses \cases{ \TH_{\la/\mu}[1;q,t\, ] & if $\mu\con \la$\cr\cr 0 & otherwise. \cr } \eqno \II.24 $$ } %%%%} We shall see that the identity in I.24 constitutes a new derivation and sharpening of the symmetric functions results of Sahi and Knop. \sas In summary, the apparently simple identity in I.16 has astonishing consequences. Several important results in the Theory of Macdonald polynomials may be derived from it. Namely, \item {(1)} We recover the plethystic formulas for the Macdonald coefficients $K_{\la\mu}(q,t)$, in a simpler and more effective form than in [7] and [8]; \item{(2)} We obtain a new and simple proof of the theorem [7], [8], [10], [11], [12], [13], [16] that the $K_{\la\mu}(q,t)$ are polynomials with integer coefficients. \item{(3)} We recover the vanishing theorem of Knop [11], [12] and Sahi [16] in a strong ``extended'' vanishing form, with an exact formula for their vanishing polynomials and a natural interpretation for their values at the points where they do not vanish. \item{(4)} Finally we shall see that the curious and remarkable Koornwinder-Macdonald reciprocity formula [15] (VI (6.6)) is but a simple specialization of I.20. \sas \noindent As we shall see the derivation of all these results is not difficult and uses no machinery other than well-known symmetric function theory. It does however depend on the discovery of certain plethystic operator identities that do provide a powerful insight into Macdonald Theory. \sas This paper is divided into 4 sections. In the first section we introduce our basic tools which consist of plethystic forms of familiar symmetric function operations and certain new plethystic operators which naturally emerge in computations involving the polynomials $\TH_\mu$. The identities we prove there should have independent interest and have been shown to have further important applications (see [2]). In Section 2 we prove Theorems I.1 -- I.4. In Section 3 we give our applications including our derivation of the Sahi-Knop symmetric function results and the reciprocity formula. Our developments rely on a number of identities for the polynomials $\TH_\mu[X;q,t]$ that may be derived from corresponding identities for the Macdonald polynomials $P_\la[x;q,t]$. The derivations that are less accessible will be carried out in Section 4, the others will be referred to the appropriate sources. \sap \heading{\bol 1. The basic tools} \sas We shall start by reviewing a few facts about Schur functions we will need in our presentation. Recall that the Littlewood-Richardson coefficients $c_{\mu\nu}^\la$ occur in the expansion $$ S_\mu \, S_\nu \ses \sum_\la c_{\mu\nu}^\la\, S_\la \ess , \eqno 1.1 $$ and in the addition formula $$ S_\la[X+Y]\ses \sum_\mu\sum_\nu c_{\mu\nu}^\la\,S_{\mu}[X] \, S_{\nu}[Y]\ess . \eqno 1.2 $$ The same coefficients are used to define the ``{\ita skew\/}'' Schur function $S_{\la/\mu}$ by setting $$ \del_{S_\mu}\, S_\la\ses S_{\la/\mu}\ses \sum_\nu c_{\mu\nu}^\la\, S_\nu \ess . \eqno 1.3 $$ In the present context we shall interpret 1.1 and 1.3 as expressing the action, on the Schur basis, of the two operators ``$\uS_\mu$'' and ``$\del_{S_\mu}$'' respectively representing ``{\ita multiplication}'' and ``{\ita skewing}'' by $S_\mu$. Note that since the orthogonality of Schur functions with respect to the Hall scalar product gives $$ \LL S_\mu\, S_\nu\scs S_\la\RR \ses c_{\mu\nu}^\la \ses \LL S_\nu\scs S_{\la/\mu} \RR\ess , \eqno 1.4 $$ we see that 1.4 may be viewed as expressing that $\del_{S_\mu}$ is the Hall scalar product adjoint of $\uS_\mu$. \sas In the same vein we can define two more general ``{\ita multiplication}'' and ``{\ita translation}'' operators ``$\PAU_Y$'' and ``$\TAU_Y$'' by setting for any given ``{\ita alphabet}'' $Y$ $\;\;(\dag)$ \FOOTNOTE{}{$(\dag)$\ess We use the word ``alphabet'' here in a very general manner, since $Y$ itself may be any algebraic expression that can be plethystically substituted into a symmetric function. For example see formulas 1.6 a) and b) below. }, and any symmetric function $Q[X]\in \LA$ $$ \eqalign{ a)\ess\ess\ess \TAU_Y\,\, Q[X] &\ses Q[X+Y]\cr b)\ess\ess\ess \PAU_Y\,\, Q[X] &\ses \OM[XY]Q[X]\ess .\cr } \eqno 1.5 $$ These operators have the following useful ``Schur function''expansions: \sas \heading{\bol Theorem 1.1} $$ \eqalign{ a)\ess\ess\ess \TAU_Y &\ses \sum_\mu\,\, S_\mu[Y]\, \del_{S_\mu}\ess .\cr b)\ess\ess\ess \PAU_Y &\ses \sum_\mu\,\, S_\mu[Y]\, {\uS_\mu}\ess .\cr } \eqno 1.6 $$ {\ita In particular we see that when $Y$ consists of a single variable $u$, we have} $$ \TAU_u\, \ses \sum_{m\geq 0}u^m \,\, \del_{S_m}\ess . \eqno 1.7 $$ \heading{\bol Proof} Note that in view of 1.3, formula 1.2 may be written in the form $$ S_\la[X+Y]\ses \sum_\nu\, S_\nu[Y]\, S_{\la/\nu}[X]\ess . $$ In other words we have $$ \TAU_Y\, S_\la[X]\ses \sum_\nu\, S_\nu[Y]\, \del_{S_\nu}\, S_{\la}[X]\ess . $$ This proves 1.6 a) when $\TAU_Y$ acts on the Schur basis. Thus the result must hold true for all symmetric functions. To prove 1.6 b) we simply observe that from the Cauchy identity we derive that for $P[X]\in \Lambda$ $$ \PAU_Y\, P[X] = \OM[XY]\, P[X] = \sum_\rho S_\rho[Y]\, S_\rho[X]\, P[X] = \Big(\sum_\rho S_\rho[Y]\, \uS_\rho[X]\Big)\, P[X]\ess . $$ Finally, we see that 1.6 a) reduces to 1.7 when $Y=\{u\}$, because $S_\rho[u]$ fails to vanish identically only when $\rho=\{m\}$. This completes our proof. \sas Our developments crucially depend on the operators $D_k$ and $D_k^*$ defined by setting for every $F\in \La$: $$\def\dsquad{\hskip6pt} \dsmatrix{ a)& D_k\, F[X] \hfill& =\; F\big[X\ssps {\TS {M \over z}}\, \big]\, \OM[-z\, X\, ]\ssp \big|_{z^k}\hfill \cr b)&D_k^*\, F[X] \hfill& =\; F\big[X\ssms {\TS {\TM \over z}}\, \big]\, \OM[z\, X\, ]\ssp \big|_{z^k}\hfill } \ess\ess (\dag) \ess\ess \hbox{for $\, -\infty {\rightarrow} %\def\LL{\big\langle} %\def \ra {\rightarrow} %\def\RR {\big\rangle} %\def\splangle{\langle\ssp } %\def\sprangle {\ssp \rangle} %\def \UU {{\scriptscriptstyle U} } %\def \HHH {{\bf H}} %\def\OO {\Omega} %\def\DD {\Delta} %\def\OM {\Omega} %\def\TOM {\tilde {\Omega}} %\def\om {\omega} %\def\la {\lambda} %\def \RA {\rightarrow} %\def \LA {\lefttarrow} %\def\xon {x_1,x_2,\ldots ,x_n} %\def\GA {\Gamma} %\def \sas {\vskip .06truein} %\def\sa{{\vskip .125truein}} %\def\msas{{\vskip -.01in}} %\def\sap{{\vskip .25truein}} %\def\sapp {{\vskip .5truein}} %\def\sss {\sigma} %\def \eee {\epsilon} %\def\DDD {\Delta} %\def\aaa {\alpha} %\def\bbb {\beta} %\def\ddd{\delta} %\def\ttt{\theta} %\def\ggg {\gamma} %\def\aa {\alpha} %\def\bb {\beta} %\def\dd{\delta} %\def\gg {\gamma} %\def\con {\subseteq} %\def \ses {\enskip = \enskip} %\def \sps {\enskip + \enskip} %\def \ssms {\, -\,\,} %\def \sses {\, =\,\,} %\def \sscs {\, ,} %\def \ssps {\,+\, \,} %\def \sms {\enskip -\enskip} %\def \lag {\langle\ssp } %\def \shhh {{\scriptstyle \cup\hskip -.18em\cup}} %\def \rag {\ssp \rangle} %\def \scs {\ssp , \ssp} %\def \ess {\enskip} %\def \ssp {\hskip .25em} %\def \bigsp {\hskip .5truein} %\def \part {\vdash} %\def \scos {\ssp :\ssp} %\def \DD {\Delta} %\def \X {{X}} %\def \Y {{Y}} %\def \Sl {S_\lambda} %\def \oom {{\omega}} %\normal %\def\uul{\underline} %\vsize=8truein %%\normalbaselineskip=12pt 18pt for space and a half, %%\normalbaselines24pt for double space %\sap %\def\today{\ifcase\month\or %January\or February\or March\or April\or may\or June\or %July\or August\or September\or October\or November\or %December\fi %\space\number\day, \number\year} % %\def \LRA {{\ssp \longrightarrow \ssp}} %\def \RA {{ \rightarrow }} %\def \LA {{ \leftarrow }} %\def \BH {{\bf H}} %\def \CC {{\cal C}} %\def \CR {{\cal R}} %\def \BV {{\bf V}} %\def \BR {{\bf R}} %\def \BQ {{\bf Q}} %\def \BM {{\bf M}} %\def \BPI {{\bf \PI'}} %\def \DA {{\downarrow}} %\def \TM {{\tilde M}} %\def \om {\omega} %\def \BH {{\bf H}} %\def \TK {{\tilde K}} %\def \Om {\Omega} %\def \bk {{\bf k}} %\def \TH {{\tilde H}} %\def \th {{\tilde h}} %\def \xon {x_1,\ldots ,x_n} %\def \yon {y_1, \ldots ,y_n} %\def \LLL {\langle} %\def \RRR {\rangle} %\def \BH {{\bf H}} %\def \BR {{\bf R}} %\def \BQ {{\bf Q}} %\def \TAU {{\cal T}} %\def \PAU {{\cal P}} %\def \uS {{\underline S}} %\def \ue {{\underline e}} %\def \LA {\Lambda} %\def \TS {\textstyle} %\def \om {\omega} %\def \TK {{\tilde K}} %\def \TH {{\tilde H}} %\def \xon {x_1,\ldots ,x_n} %\def \yon {y_1, \ldots ,y_n} %\def \LLL {\langle} %\def \RRR {\rangle} %\def \xom {x_1,\ldots ,x_m} %\def \sgmu {\ ^{sg}\BH_\mu} %\def \bhmu {\BH_\mu} %\def \pmu {\rho_\mu} %\def \FP {{\cal FP}[X,Y]} %\def \gamu {{\bf \Gamma}_\mu} %\def \mgamu {m({\bf \Gamma}_\mu)} %\def \RImu {{\cal RI}_\mu} %\def \da {\downarrow} %\def \sda {\hskip -.03truein\downarrow} %\def \dan {\hskip -.03truein\downarrow_n} %\def \ua {\uparrow} %\def \sua {\hskip -.03truein\uparrow} %\def \os {\overline {s}} %\def \ox {\overline {x}} %\def \uan {\hskip -.03truein\uparrow^n} %\def\today{\ifcase\month\or %January\or February\or March\or April\or May\or June\or %July\or August\or September\or October\or November\or %December\fi %\space\number\day, \number\year} %\headline={\small (Preliminary Version)\hfill\small Explicit formulas for q,t -Kotska $\ess\ess\ess\ess\ess$ %\hfill$\ess\ess\ess$\today$\ess\ess\ess\ess\ess\ess$ \folio } \footline={\hfil} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sa \heading{\bol 2. Proofs of the main results.} Our arguments here hinge on the following fundamental fact: \sas \heading{\bol Theorem 2.1} {\ita Every symmetric polynomial $P$, homogeneous of degree $k\geq 1$, may be written in the form $$ P\ses D_1\, A\sps \ue_1\, B \eqno 2.1 $$ with $A,B $ homogeneous symmetric polynomials of degree $k-1$. Moreover, if $P\in \LA_{Z[q,t]}$ then 2.1 holds true with $A=R[X;q,t\, ]/M^{k-1}$ and $B=S[X;q,t\, ]/M^{k-1} ,$ with $R$ and $S$ polynomials in $\LA_{Z[q,t]}$. Of course, the same result holds true with $D_1$ replaced by $D_1^*$ in 2.1. } \noindent {\bol Proof} It is sufficient to work with $D_1$ since the result for $D_1^*$ immediately follows by an application of 1.14~c). For convenience, we shall write $$ U[X;q,t\, ]\ssp \equiv _{\,\, \ue_1}\ssp V[X;q,t\, ] \eqno 2.2 $$ to indicate that $\,\, U[X;q,t\, ]-V[X;q,t\, ]=e_1\,S[X;q,t\, ] $ with $S[X;q,t\, ]\in \LA_{Z[q,t\, ]}$. This given, we shall show that for every elementary basis element $e_\aaa=e_{\aaa_1}e_{\aaa_2}\cdots e_{\aaa_m}$ with $\aaa\part k$, we have an identity of the form $$ M^{k-1}\, e_\aaa=M^{k-1}\, e_{\aaa_1}e_{\aaa_2}\cdots e_{\aaa_m} \ssp \equiv _{\,\, \ue_1}\ssp D_1\, R[X;q,t\, ] \eqno 2.3 $$ (with $R[X;q,t\, ]\in \LA_{Z[q,t\, ]}$\ssp) . We shall prove 2.3 by a process which was first used in [7]. The idea is to proceed by an induction which ``descends`` on the largest part of $\aaa$. To begin with note that we have $$ \eqalign{ D_1\, e_1^{k-1}&\ses \big(e_1\big[X+{\textstyle{M\over z}}\big]\big)^{k-1}\OM[-zX]\, \Big|_z\cr & \ses \big(e_1[X]+{\textstyle{M\over z}}\big)^{k-1}\OM[-zX]\, \Big|_z\cr &\ess \equiv _{\,\, \ue_1} {M^{k-1}\over z^{k-1}}\,\, \OM[-zX]\, \Big|_z\ses (-1)^k\, M^{k-1} e_k[X]\ess . \cr } $$ In other words $$ M^{k-1}\, e_k[X] \ess \equiv _{\,\, \ue_1} \ess (-1)^k \ssp D_1\, e_1^{k-1}\ess . \eqno 2.4 $$ This proves 2.3 when the largest part of $\aaa$ is as large as possible. Let us then assume that we have $$ M^{k-1}\, e_{\bbb_1}e_{\bbb_2}\cdots e_{\bbb_m}\;\equiv _{\,\, \ue_1}\; D_1\, R_\bbb[X;q,t\,] \ess\ess (\hbox{with $R_\bbb[X;q,t]\in \LA_{Z[q,t ]}$}) \eqno 2.5 $$ when $\,\, \bbb_1>a$. Our goal is to use this to prove 2.3 for $$ \aaa=(a\geq \aaa_2\geq \cdots \geq \aaa_k\geq 0)\ess . $$ To this end we note that $$ \eqalign{ D_1\, e_1^{a-1}\, e_{\aaa_2}\cdots e_{\aaa_k} &=\; \big(e_1+{\textstyle{M\over z}}\big)^{a-1}\, \left(\, \prod_{i=2}^k\Big(\sum_{r_i=0}^{\aaa_i}e_{\aaa_i-r_i}[X]\, {\textstyle{ e_{r_i}[M]\over z^{r_i}}}\Big)\right)\, \OM[-zX]\, \Big|_{z} \cr &\equiv _{\,\, \ue_1}\ssp {M^{a-1}\over z^{a }} \left(\, \prod_{i=2}^k\Big(\sum_{r_i=0}^{\aaa_i}e_{\aaa_i-r_i}[X]\, {\textstyle{ e_{r_i}[M]\over z^{r_i}}}\Big)\right)\, \OM[-zX]\, \Big|_{z^0}\ess . \cr } $$ Multiplying both sides by $M^{k-a}$, expanding the product and arranging the resulting terms according to increasing powers of $z$ gives $$ \dsmatrix{ M^{k-a}\, D_1\, e_1^{a-1}\, e_{\aaa_2}\cdots e_{\aaa_k}\ssp \hfill\cr\hskip .3in %& \equiv _{\,\, \ue_1} M^{k-1}\bigg({e_{\aaa_2}\cdots e_{\aaa_k}\over z^a}+ \sum_{b>a} \ssp \sum_{\bbb_2\geq \cdots \geq\bbb_k}c_{b, \bbb_2, \cdots , \bbb_k }[M] \ssp {e_{\bbb_2}\cdots e_{\bbb_k}\over z^b}\bigg)\OM[-zX]\, \Big|_{z^0} \hfill\cr\hskip.3in %& \equiv _{\,\, \ue_1} M^{k-1} \bigg( (-1)^a\, e_a e_{\aaa_2}\cdots e_{\aaa_k} + \sum_{b>a}(-1)^b \;\sum_{\hbox to0pt{\hss$\scriptstyle\bbb_2\geq \cdots \geq\bbb_k$\hss} } c_{b, \bbb_2, \cdots , \bbb_k }[M] \ssp e_{b}\, e_{\bbb_2}\cdots e_{\bbb_k} \bigg) \hfill\cr }$$ with $c_{b, \bbb_2, \cdots , \bbb_k }[M]$ an elementary basis element plethystically evaluated at $M$. In other words we obtain that $$\eqalign{ M^{k-1} e_a e_{\aaa_2}\cdots e_{\aaa_k} &\equiv _{\,\, \ue_1} { (-1)^a M^{k-a}} D_1\, e_1^{a-1}\, e_{\aaa_2}\cdots e_{\aaa_k} \cr&\hskip.4in\sms \sum_{b>a}(-1)^{b-a} \hskip- .16in \sum_{\bbb_2\geq \cdots \geq\bbb_k }c_{b, \bbb_2, \cdots , \bbb_k }[M] \ssp M^{k-1} e_{b}e_{\bbb_2}\cdots e_{\bbb_k} } $$ and the induction hypothesis in 2.5 yields 2.3 as desired. \sa We are now in a position to give our \sas \heading{\bol Proof of Theorem I.2} We shall begin by showing I.16. To this end it will be convenient to write our operator $\BPI$ in the form (see I.15) $$ \BPI\ses \nabla^{-1}\TAU_\eee^{-1}\bigsp\bigsp (\hbox{with $\eee=^-1$}). \eqno 2.6 $$ This given, we are to show that for any homogeneous polynomial $f$, of degree $d(f)$, we have $$ \LL f\scs \TH_\mu[X+1;q,t\,]\RR_*\ses\BPI_f\big[D_\mu\big]\bigsp\bigsp (\hbox{for all $\mu$}). \eqno 2.7 $$ We shall proceed by induction on $d(f)$. Since $\TH_\emptyset=1$ we have $$ \nabla\, 1\ses \nabla^{-1}\, 1\ses 1\ess . \eqno 2.8 $$ Thus 2.6 gives $\BPI_f\equiv 1$ for $f\equiv 1$. On the other hand, the expansion in I.5 and the normalization in 1.12 yield that $$ \LL 1\scs \TH_\mu[X+1;q,t\,]\RR_*\ses\TH_\mu[1;q,t\,]\ses \TK_{n,\mu}(q,t)\ses \TH_\mu \, \big|_{S_n}\ses 1\ess . $$ Thus 2.7 is trivially true when $f$ is a constant, and we can start our induction at $d(f)=0$. Let us then assume that 2.7 is true for all $\mu\part n$ and for $d(f)0$\ssp , \cr\cr B_\mu(q,t) & if $r=0$\ssp . \cr } \eqno 2.14 $$ This given, we see that 2.13 for $k=1$ and $B=1$ reduces to $$ \LL e_1 \scs \TH_\mu[X+1;q,t\,]\RR_* \ses M\, \sum_{\nu\RA\mu}c_{\mu\nu}(q,t)\ses M B_\mu(q,t)\ses D_\mu(q,t)+1 $$ Thus the validity of 2.7 for the case $f=e_1\cdot 1$ requires that $$ \nabla^{-1}\TAU_{\eee}^{-1}e_1 \ses e_1+1\ess . \eqno 2.15 $$ To verify this, we note that definition in 1.8 a), 2.8 and 2.11 give $$ -e_1=\OM[-zX]\big|_{z}\ses D_1\cdot 1 \ses D_1 \nabla\, 1\ses -\nabla e_1\cdot 1\ses -\nabla e_1\ess . $$ In other words $$ \nabla^{-1}e_1\ses e_1\ess . $$ We have then $$ \nabla^{-1}\TAU_\eee^{-1}e_1\ses \nabla^{-1}(e_1-e_1[\eee])\ses \nabla^{-1}(e_1+1)\ses e_1+1\ssp , $$ as desired. This establishes Case 2) for $k=1$. Let us now deal with the case when $B$ is of degree $k-1>0$. To this end, we start by using the induction hypothesis in 2.13 and get $$ \LL e_1 B\scs \TH_\mu[X+1;q,t\, ]\RR_* \ses M \sum_{\nu\RA \mu}c_{\mu\nu}(q,t)\, \BPI_B\big[ D_\nu(q,t)\big]\ess . \eqno 2.16 $$ Since $D_\nu=D_\mu-M T_{\mu/\nu}$, by I.4 and 1.33 we may write $$ \BPI_B\big[ D_\nu(q,t)\big]\ses \sum_{r= 0}^{k-1}\BPI_B\big[ D_\mu-M/z\big]\big|_{z^{-r}} (T_{\mu/\nu})^r\ess . $$ Substituting this back into 2.16 gives $$ \eqalign{ \LL e_1 B\,, \TH_\mu[X+1;q,t ]\RR_* &= M \sum_{\nu\RA \mu}c_{\mu\nu}(q,t)\, \sum_{r= 0}^{k-1}\BPI_B\big[ D_\mu-M/z\big]\big|_{z^{-r}} (T_{\mu/\nu})^r\cr &= M \sum_{r= 0}^{k-1}\BPI_B\big[ D_\mu-M/z\big]\big|_{z^{-r}} \sum_{\nu\RA \mu}c_{\mu\nu}(q,t)\, (T_{\mu/\nu})^r\, . \cr }\hskip -.3in \eqno 2.17 $$ We now use 2.14 and get $$ \eqalign{ \LL e_1 B \,,\, \TH_\mu[X+1;q,t\, ]\RR_* \hskip-1in\cr &=\, M\, \BPI_B\big[ D_\mu\big] B_\mu(q,t) \cr & \hskip.5in \sps M \sum_{r= 1}^{k-1} \BPI_B\big[ D_\mu-M/z\big]\big|_{z^{-r}}\times{t\, q\over M}\, h_{r+1}\big[D_\mu(q,t)/t q\big] \cr &=\, M\, \BPI_B\big[ D_\mu\big] B_\mu(q,t)\sms \BPI_B\big[ D_\mu\big] D_\mu(q,t) \cr & \hskip .5in \sps \sum_{r= 0}^{k-1} \BPI_B\big[ D_\mu-M/z\big]\big|_{z^{-r}}\times {t\, q}\,\OM\big[zD_\mu(q,t)/t q\big]\big|_{z^{r+1}} \cr &=\, \BPI_B\big[ D_\mu\big] + \sum_{r= 0}^{k-1} \BPI_B\big[ D_\mu-M/z\big]\big|_{z^{-r}}\times{t\, q}\,\OM\big[zD_\mu(q,t)/t q\big]\big|_{z^{r+1}}\ssp . \cr} $$ In other words we have $$ \LL e_1 B\,,\, \TH_\mu[X+1;q,t]\RR_* =\, \BPI_B\big[ D_\mu\big] \,+\, \BPI_B\big[ D_\mu-M/tq z\big] \OM\big[zD_\mu(q,t)\big]\Big|_{z} \, . \; \eqno 2.18 $$ This equality results from the fact that for any two formal power series $\Phi(z),\Psi(z)$, we have $$ \sum_{r\geq 0}\Phi(1/z)\Big|_{z^{-r}}\times q\, t\, \Psi(z/q\, t)\big|_{z^{r+1}}\,=\, q\, t\, \Phi(1/z)\Psi(z/q\, t)\big|_{z} \,=\, \Phi(1/tqz)\Psi(z)\big|_{z}\, . $$ Recalling the definition of $D_1^*$ given in 1.8, we see that 2.18 is none other than $$ \LL e_1 B\scs \TH_\mu[X+1;q,t\, ]\RR_*\ses\BPI_B\big[ D_\mu\big] \sps (D_1^*\BPI_B)\big[ D_\mu\big]\ess . $$ Thus the validity of 2.7 for $f=e_1B$ is reduced to showing that we have $$ \BPI_{e_1B}\ses \BPI_B \sps D_1^*\BPI_B \ess . \eqno 2.19 $$ For this to hold for all $k$ we must have $$ \nabla^{-1}\TAU_\eee^{-1}\ue_1\ses \nabla^{-1}\TAU_\eee^{-1}\sps D_1^*\, \nabla^{-1}\TAU_\eee^{-1}\ess , $$ or better, multiplying on the right by $\TAU_\eee\nabla$, $$ \nabla^{-1}\TAU_\eee^{-1}\ue_1\,\TAU_\eee\nabla\ses {1}\sps D_1^*\ess . \eqno 2.20 $$ Now to show this, note that for any polynomial $P[X]$ we have $$ \TAU_\eee^{-1}\ue_1\,\TAU_\eee \, P[X]\ses \TAU_\eee^{-1} \ue_1 \, P[X+\eee\,]\ses (\ue_1-e_1[\eee])P[X]\ses (\ue_1+1)P[X]\ess . $$ Thus 2.20 is equivalent to $$ \nabla^{-1}(\ue_1+1)\nabla\ses { 1}\sps D_1^*\ess . $$ Namely $$ \nabla^{-1} \ue_1 \nabla\ses \, D_1^*\ess . \eqno 2.21 $$ But this is none other than what we obtain by equating the right hand side of 1.28 b) to the right hand side of 1.30 b*). This completes the proof of I.16. Note then that when $d(f)=k\leq n$ we can use 1.45 b) and obtain that $$ \LLL f\scs \TH_\mu[X+1;q,t\, ]\RR_*\ses \LL e_{n-k}^*f\scs \TH_\mu\RR_* \eqno 2.22 $$ which yields I.18 a). As for I.18 b), we note that 2.22 for $f\, \RA\, f[{-^-X\over M}]=\om\phi^{-1} f $ gives $$ \eqalign{ \LLL \om \phi^{-1}f\scs \TH_\mu[X+1;q,t\, ]\RR_* &\ses \LL (\om\phi^{-1} h_{n-k} )\, \om\phi^{-1} f\scs \TH_\mu\RR_* \cr&\ses \LL \om\phi^{-1} (h_{n-k} f)\scs \TH_\mu\RR_* \ses \LL h_{n-k}\, f \scs \TH_\mu\RR \ess , } $$ and I.18 a) gives I.18 b) with $$ {\bf \PI}_f[X;q,t\,]\ses \BPI_{\om\phi^{-1} f}[X;q,t\,]\ses \nabla^{-1}\TAU_\eee^{-1} f\big[{\textstyle {-^-X\over M}}\big] \ses\nabla^{-1}f\big[{\textstyle{1-^-X\over (1-t)(1-q) }}\big] $$ completing the proof of Theorem I.2. \sa \heading{\bol Proof of Theorem I.3} For convenience let us set $$ E_\mu[X;q,t\, ]\ses \OM[{\textstyle{X\over M}}] \,\nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu\over M}}X]\ess . \eqno 2.23 $$ This given, the equality $$ E_\mu[X;q,t\, ]=\TH_\mu[X+1;q,t\,] \eqno 2.24 $$ is a consequence of the following remarkable sequence of equivalent expressions: $$\hskip-2pt \def\dsquad{\hskip 6pt} \dsmatrix{ \hfill \BPI_f [D_\mu\, ;q,t] &=& \LL \nabla^{-1}\, f\, [X-^-1;q,t]\sscs \OM[D_\mu X]\RR \hfill&(1)\cr % \hfill \hbox{\rm (1.40)}\RA\; &=& \LL \nabla^{-1}\, f\, [X-^-1;q,t]\sscs \om\, \OM[{\textstyle{D_\mu \over M}}X]\RR_* \hfill&(2)\cr % \hfill \hbox{\rm(Prop.\ 1.9)}\RA\; &=& \LL f\, [X-^-1;q,t]\sscs \nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu \over M}}X]\RR_* \hfill&(3)\cr % \hfill \hbox{\rm (def.\ 1.5a) \& 1.37)}\RA\; &=& \LL \TAU_{-^-1} f\, [X;q,t]\sscs \phi\, \om\, \nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu \over M}}X]\RR \hfill&(4)\cr % \hfill \hbox{\rm (def.\ 1.5$\,$b)$\,$)}\RA\; &=& \LL f\, [X;q,t]\sscs \PAU_{-^-1}\, \phi\, \om\, \nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu\over M}}X]\RR \hfill&(5)\cr % \hfill \hbox{\rm (I.2 \& def.\ 1.38)}\RA\; &=& \Big\langle f\, [X;q,t]\sscs \phi\, \om\,\Big( \OM[{\textstyle{X\over M}}] \,\nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu\over M}}X]\Big)\Big\rangle \hfill&(6)\cr % \hfill \hbox{\rm (1.37)}\RA\; &=& \LL f\, [X;q,t]\sscs \OM[{\textstyle{X\over M}}] \,\nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu\over M}}X] \RR_* \ess . \hfill&(7) } \eqno 2.25 $$ In other words, we must have $$ \BPI_f [D_\mu\, ;q,t]\ses \LL f\, \sscs E_\mu[X;q,t]\RR_*\ess . $$ Combining this with 2.7 we obtain that $$ \LL f \sscs \, \TH_\mu[X+1]\, \RR_*\ses \LL f \sscs E_\mu[X;q,t]\RR_* \eqno 2.26 $$ must hold true for any symmetric polynomial $f$, forcing the equality in 2.24. This completes our proof of Theorem I.3. \sas \heading{\bol Remark 2.1} We should note that, conversely, starting from 2.24 and carrying out the steps in 2.25 in the reverse order $(7)\RA(6) \RA\cdots\RA (2)\RA(1)$ shows that I.20 and I.16 are simply equivalent statements. \sas \sas \heading{\bol Remark 2.2} We should also mention that the identity in I.16 contains the remarkable fact that for any homogeneous polynomial $P$ of degree $d(P)$ we have $$ \BPI_P[D_\mu(q,t)\,]\ses 0\bigsp\ess\ess\hbox{for all $|\mu|0)$ we have $$ S_\la[X]\ses \BH_{\la_1}\BH_{\la_2}\cdots \BH_{\la_k}\, { 1}\ess . \eqno 2.30 $$ This is a classical result whose proof may be found in Section 4. This given we see that when $\la$ is of the form $(n-k,\ggg)$ with $\ggg\part k$, we can write $$ \eqalign{ S_{(n-k,\ggg)}[X]&\ses \BH_{n-k}\, S_\ggg[X]\ses S_\ggg[X-{\textstyle{1\over z}}]\, \OM[zX]\big|_{z^{n-k}}\cr &\ses\sum_{i=0}^{|\ggg|}S_\ggg[X-{\textstyle{1\over z}}]\, \big|_{z^{-i}}\,\,\OM[zX]\big|_{z^{n-k+i}}\cr &\ses \sum_{i=0}^{|\ggg|}\, h_{n-k+i}[X]\, S_\ggg[X-{\textstyle{1\over z}}]\, \big|_{z^{-i}}\ess . \cr } $$ In particular, we derive that $$ S_{(n-k,\ggg)'}^*[X]\ses \sum_{i=0}^{|\ggg|}\, e_{n-k+i}^*[X]\, S_\ggg[{\textstyle{X\over M}}-{\textstyle{1\over z}}]\, \big|_{z^{-i}} \ess . $$ Substituting this in 2.28 gives $$ \TK_{(n-k,\ggg),\mu}(q,t)\ses \sum_{i=0}^{|\ggg|}\, \LL e_{n-k+i}^*[X]\, S_\ggg[{\textstyle{X\over M}}-{\textstyle{1\over z}}] \scs \TH_\mu\RR_*\, \Big|_{z^{-i}}\ess . \eqno 2.31 $$ From I.18 a) we then get that $$ \LL e_{n-k+i}^*[X]\, S_\ggg[{\textstyle{X\over M}}-{\textstyle{1\over z}}] \scs \TH_\mu\RR_*\, \Big|_{z^{-i}}\ses \BPI_{P_i}[D_\mu(q,t)\,] $$ with $$ P_i\ses S_\ggg[{\textstyle{X\over M}}-{\textstyle{1\over z}}]\, \Big|_{z^{-i}}\ess . $$ Thus $$ \TK_{(n-k,\ggg),\mu}(q,t)\ses \sum_{i=0}^{|\ggg|}\,\BPI_{P_i}[D_\mu(q,t)\,]\ess . \eqno 2.32 $$ Now the definition in 2.6 gives $$ \BPI_{P_i}\ses \nabla^{-1}S_\ggg[{\textstyle{X-^-1\over M}}-{\textstyle{1\over z}}]\, \Big|_{z^{-i}}\ess . $$ Substituting this in 2.32, we finally get $$ \eqalign{ \TK_{(n-k,\ggg),\mu}(q,t)&\ses \sum_{i=0}^{|\ggg|}\, \nabla^{-1}S_\ggg[{\textstyle{X-^-1\over M}}-{\textstyle{1\over z}}]\, \Big|_{z^{-i}}\ess\bigg|_{X\RA D_\mu(q,t)} \cr &\ses \nabla^{-1}S_\ggg[{\textstyle{X-^-1\over M}}-1]\ess\bigg|_{X\RA D_\mu(q,t)}\ess . \cr } $$ This completes our proof of Theorem I.1. \sas \heading{\bol Remark 2.3} It follows from Theorem I.3 that the expression $$ E_\mu[X;q,t\, ]\ses \OM[{\textstyle{X\over M}}] \,\nabla^{-1}\,\om\, \OM[{\textstyle{D_\mu\over M}}X]\ess \eqno 2.33 $$ defines a polynomial of degree $|\mu|$. However, there is a more illuminating way to see this. To begin with, note that from 1.18 c) and the definition in I.8, we get that $$ \nabla^{-1}\om\OM[{\textstyle{D_\mu\over M}}X] \ses \sum_\aaa\, {\TH_\aaa[X;q,t\, ]\over T_\aaa\,}\, { \TH_\aaa[D_\mu]\over \th_\aaa\th_\aaa ' }\ess . \eqno 2.34 $$ On the other hand, since we have $$ \OM[{\textstyle{X\over M}}] \ses \tilde \OM[{\textstyle{-^-X\over M}}] \ess , $$ again from 1.18 c) we obtain that$$ \OM[{\textstyle{X\over M}}] \ses \sum_\bbb\, \TH_\bbb[X;q,t\, ]\, { \TH_\bbb[-^-1]\over \th_\bbb\th_\bbb' }\ess . \eqno 2.35 $$ Multiplying 2.35 and 2.34, the definition in 2.33 gives $$ E_\mu[X;q,t\, ] \ses \sum_\aaa\,\sum_\bbb\, \TH_\aaa[X ]\,\TH_\bbb[X ]\, { \TH_\aaa[D_\mu]\over T_\aaa\,\th_\aaa\th_\aaa ' }\, { \TH_\bbb[-^-1]\over \th_\bbb\th_\bbb' }\ess . \eqno 2.36 $$ We can thus apply 1.46 and 1.49, and then 1.48 to obtain $$ \eqalign{ E_\mu[X;q,t\, ] &\ses \sum_\aaa \sum_\bbb\, \Big(\sum_{\la\supseteq \, \aaa , \bbb}\, d_{\aa,\bbb}^\la\, \TH_\la[X]\Big) { \TH_\aaa[D_\mu]\over T_\aaa\,\th_\aaa\th_\aaa ' }\, { \TH_\bbb[-^-1]\over \th_\bbb\th_\bbb' } \cr &\ses \sum_{\la}\,\TH_\la[X]\, \sum_{\aaa , \bbb\con \la}\, \, d_{\aa,\bbb}^\la\,{ \TH_\aaa[D_\mu]\over T_\aaa\,\th_\aaa\th_\aaa ' }\, { \TH_\bbb[-^-1]\over \th_\bbb\th_\bbb' } \cr &\ses \sum_{\la}\,{\TH_\la[X]\over\th_\la\th_\la ' }\, \Big( \sum_{\aaa , \bbb\con \la}\, \, {c_{\aa,\bbb}^\la\over T_\aaa\,}\,{ \TH_\aaa[D_\mu] }\, { \TH_\bbb[-^-1] }\Big) \cr &\ses \sum_{\la}\,{\TH_\la[X]\over\th_\la\th_\la ' }\, \Big( \sum_{\aaa , \bbb\con \la}\, \, {c_{\aa,\bbb}^\la\over T_\aaa\,}\,{ \TH_\aaa[Y] }\, { \TH_\bbb[-^-1] }\Big)\ssp \Big|_{Y\RA D_\mu} \cr &\ses \sum_{\la}\, \TH_\la[X] \ess {\nabla_y^{-1}\,\TH_\la[Y-^-1]\over\th_\la\th_\la ' }\ssp \Big|_{Y\RA D_\mu} \ses \sum_{\la}\, \TH_\la[X] \ess \delta_\la\big [D_\mu \big]\ess , \cr } $$ which shows that the polynomiality of $E_\mu[X;q,t\,]$ is a direct consequence of the vanishing properties of $\delta_\la$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\hsize 6truein %\def\del {\partial} %\def \con {\subseteq} %\def \scong {\ess \cong \ess } %\def \brkpg {\supereject} %\def\lll{\lambda} %\def\La {\Lambda} %\def\sig{\sigma} %\def\dia{\diamondsuit} %\def \ggg {\gamma} %\def \GG {\Gamma} %\def\PI{\Pi} %\def \-> {\rightarrow} %\def\LL{\big\langle} %\def \ra {\rightarrow} %\def\RR {\big\rangle} %\def\splangle{\langle\ssp } %\def\sprangle {\ssp \rangle} %\def \UU {{\scriptscriptstyle U} } %\def \HHH {{\bf H}} %\def\OO {\Omega} %\def\DD {\Delta} %\def\OM {\Omega} %\def\TOM {\tilde {\Omega}} %\def\om {\omega} %\def\la {\lambda} %\def \RA {\rightarrow} %\def \LA {\lefttarrow} %\def\xon {x_1,x_2,\ldots ,x_n} %\def\GA {\Gamma} %\def \sas {\vskip .06truein} %\def\sa{{\vskip .125truein}} %\def\msas{{\vskip -.01in}} %\def\sap{{\vskip .25truein}} %\def\sapp {{\vskip .5truein}} %\def\sss {\sigma} %\def \eee {\epsilon} %\def\DDD {\Delta} %\def\aaa {\alpha} %\def\bbb {\beta} %\def\ddd{\delta} %\def\ttt{\theta} %\def\ggg {\gamma} %\def\aa {\alpha} %\def\bb {\beta} %\def\dd{\delta} %\def\gg {\gamma} %\def\con {\subseteq} %\def \ses {\enskip = \enskip} %\def \sps {\enskip + \enskip} %\def \ssms {\, -\,\,} %\def \sses {\, =\,\,} %\def \sscs {\, ,} %\def \ssps {\,+\, \,} %\def \sms {\enskip -\enskip} %\def \lag {\langle\ssp } %\def \shhh {{\scriptstyle \cup\hskip -.18em\cup}} %\def \rag {\ssp \rangle} %\def \scs {\ssp , \ssp} %\def \ess {\enskip} %\def \ssp {\hskip .25em} %\def \bigsp {\hskip .5truein} %\def \part {\vdash} %\def \scos {\ssp :\ssp} %\def \DD {\Delta} %\def \X {{X}} %\def \Y {{Y}} %\def \Sl {S_\lambda} %\def \oom {{\omega}} %\normal %\def\uul{\underline} %\vsize=8truein %%\normalbaselineskip=12pt 18pt for space and a half, %%\normalbaselines24pt for double space %\sap %\def\today{\ifcase\month\or %January\or February\or March\or April\or may\or June\or %July\or August\or September\or October\or November\or %December\fi %\space\number\day, \number\year} % %\def \LRA {{\ssp \longrightarrow \ssp}} %\def \RA {{ \rightarrow }} %\def \LA {{ \leftarrow }} %\def \BH {{\bf H}} %\def \CC {{\cal C}} %\def \CR {{\cal R}} %\def \BV {{\bf V}} %\def \BR {{\bf R}} %\def \BQ {{\bf Q}} %\def \BM {{\bf M}} %\def \BPI {{\bf \PI'}} %\def \DA {{\downarrow}} %\def \TM {{\tilde M}} %\def \om {\omega} %\def \BH {{\bf H}} %\def \TK {{\tilde K}} %\def \Om {\Omega} %\def \bk {{\bf k}} %\def \TH {{\tilde H}} %\def \th {{\tilde h}} %\def \xon {x_1,\ldots ,x_n} %\def \yon {y_1, \ldots ,y_n} %\def \LLL {\langle} %\def \RRR {\rangle} %\def \BH {{\bf H}} %\def \BR {{\bf R}} %\def \BQ {{\bf Q}} %\def \TAU {{\cal T}} %\def \PAU {{\cal P}} %\def \uS {{\underline S}} %\def \ue {{\underline e}} %\def \LA {\Lambda} %\def \TS {\textstyle} %\def \om {\omega} %\def \TK {{\tilde K}} %\def \TH {{\tilde H}} %\def \xon {x_1,\ldots ,x_n} %\def \yon {y_1, \ldots ,y_n} %\def \LLL {\langle} %\def \RRR {\rangle} %\def \xom {x_1,\ldots ,x_m} %\def \sgmu {\ ^{sg}\BH_\mu} %\def \bhmu {\BH_\mu} %\def \pmu {\rho_\mu} %\def \FP {{\cal FP}[X,Y]} %\def \gamu {{\bf \Gamma}_\mu} %\def \mgamu {m({\bf \Gamma}_\mu)} %\def \RImu {{\cal RI}_\mu} %\def \da {\downarrow} %\def \sda {\hskip -.03truein\downarrow} %\def \dan {\hskip -.03truein\downarrow_n} %\def \ua {\uparrow} %\def \sua {\hskip -.03truein\uparrow} %\def \os {\overline {s}} %\def \ox {\overline {x}} %\def \uan {\hskip -.03truein\uparrow^n} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \heading{\bol Remark 2.4} Soon after the original conjecture of formula 2.24, we discovered the following extremely simple ``{\ita proof\/}''. First rewrite the formula as $$ \TH_\mu[X;q,t\,]\ses \TAU_{-1}\PAU_{1/M} \nabla^{-1}\om\OM\big[{\textstyle{X D_\mu\over M}}\big] \ess . \eqno 2.37 $$ Now $\TH_\mu$ is uniquely characterized up to a scalar factor as the eigenfunction of $D_0$ with eigenvalue $D_\mu$, so we must verify that the right hand side of 2.37 has the same property. Now we have $$ \dsmatrix{ D_0 &=& \TAU_1^{-1}(D_0-D_{-1})\TAU_1 \hfill &\quad\quad&(\hbox {by 1.36 a)}) \hfill \cr &=& -\ssp\TAU_1^{-1} \PAU_{1/M}D_{-1}\PAU_{-1/M}\TAU_1 \hfill & &(\hbox {by 1.35 a)}) \hfill \cr &=&-M\ssp\TAU_1^{-1} \PAU_{1/M} \nabla^{-1}\del_1\nabla \PAU_{-1/M}\TAU_1 \hfill & &(\hbox {by 1.24 a) \& 1.30 a)}) \hfill } $$ and this immediately yields $$ \eqalign{ D_0\ssp \TAU_{-1}\PAU_{1/M} \nabla^{-1}\om\OM\big[{\textstyle{X D_\mu\over M}}\big] \hskip-.75in\cr &\ses -M\ssp\TAU_1^{-1} \PAU_{1/M} \nabla^{-1}\del_1\nabla \PAU_{-1/M}\TAU_1\ess \TAU_{-1}\PAU_{1/M} \nabla^{-1}\om\OM\big[{\textstyle{X D_\mu\over M}}\big] \cr &\ses -M\ssp\TAU_1^{-1} \PAU_{1/M} \nabla^{-1}\del_1\om\OM\big[{\textstyle{X D_\mu\over M}}\big] \cr {(\dag)} &\ses -M\ssp\TAU_1^{-1} \PAU_{1/M} \nabla^{-1}{D_\mu\over M}\,\,\om\OM\big[{\textstyle{X D_\mu\over M}}\big] \cr &\ses -D_\mu\,\,\TAU_1^{-1} \PAU_{1/M} \nabla^{-1}\,\om\OM\big[{\textstyle{X D_\mu\over M}}\big] \cr} $$ \FOOTNOTE{}{$(\dag)\ess$ This is because for any $A$ we have $\del_1\om \,{\rm exp}(p_1[XA])=A\,\om \,{\rm exp}(p_1[XA])\ssp . $}% as desired. The missing scalar factor is easily shown to be $1$ by setting $X=0$ in (2.23). For this proof we don't need the deeper identity 2.14; we only need that $\del_1\TH_\mu=\sum_{\nu\RA\mu}c_{\mu\nu}\TH_\nu$ without explicit knowledge of the coefficients $c_{\mu\nu}$, (that is only formula (6.7) p.~332 of [15]) . On further reflection, however, it is clear that something must be wrong with this argument. Indeed, if $A$ is any quantity whatsoever, it appears to show that the expression $$ \TAU_1^{-1}\PAU_{1/M}\nabla^{-1}\om \OM\big[{\textstyle{X A\over M}}\big] $$ is an eigenfunction of $D_0$ with eigenvalue $A$, a highly unlikely possibility! The problem is that the right hand side of 2.33 is, {\ita a priori}, a formal series, containing terms of unbounded degree. One cannot apply the operator $\TAU^{-1}$ to such a series, just as one cannot substitute $x-1$ for $x$ in a formal power series in one variable. It is possible to evade this difficulty to a certain extent by proving instead that $E_\mu[X;q,t\, ]$ is an eigenfunction of the operator $\TAU_1 D_0\TAU_1^{-1}$. This makes sense because the latter operator is equal to $D_0-D_{-1}$ which {\bol can} be applied to a formal series. The problem this causes is that if we admit formal series as eigenfunctions, the inhomogeneous operator $D_0-D_{-1}$ no longer has a ``discrete spectrum'': it has in fact infinitely many independent eigenfunctions with any given eigenvalue $A$. All we can say is that $\TH_\mu[X+1;q,t\, ]$ is its unique ``{\ita polynomial\/}'' eigenvector with eigenvalue $-D_\mu$. Absent a separate $(\dag\dag)$ \FOOTNOTE{}{$(\dag\dag)$\ess One that does not use Theorem I.2 .} demonstration that $E_\mu[X;q,t\, ]$ is a polynomial, this particular would-be ''{\ita proof\/}'' is incomplete. \sa %%%%\vbox{ \heading{\bol 3. Some applications} Formula I.11 yields yet one more path for establishing the integrality of the Macdonald $q,t$-Kostka coefficients. To see this we need a few preliminary observations. To begin with it follows from the Macdonald ``duality'' result ([15] (5.1) p.~327) that we have $$ \TK_{\la'\mu}(q,t)\ses t^{n(\mu)}q^{n(\mu')}\TK_{\la\mu}(1/q,1/t)\ess . \eqno 3.1 $$ In particular we see that if $\TK_{\la\mu}(q,t)$ is a polynomial then it must be of degrees $\leq n(\mu)$ in $t$ and $\leq n(\mu')$ in $q$. Thus the definition $\TK_{\la\mu}(q,t)= t^{n(\mu)}K_{\la\mu}(q,1/t)$ guarantees that %%%%} $$ a)\ess\ess \TK_{\la\mu}(q,t)\in {\bf Z}[q,t]\ess\ess\ess \Longleftrightarrow\ess\ess\ess b)\ess\ess K_{\la\mu}(q,t)\in {\bf Z}[q,t]\ess . \eqno 3.2 $$ There are a number of algorithms for constructing the $K_{\la\mu}(q,t)$ that stem from the various identities established in Macdonald's original papers [14], [15]. All of these algorithms introduce denominators of one kind or another. The simplest and most remarkable of these algorithms is one discovered by Vinet-Lapointe [13]. They observed that the Macdonald ``integral form'' $$ J_\mu[X_n;q,t]=\sum_{\la\part n}\, S_\la[X_n(1-t)] K_{\la\mu}(q,t) \eqno 3.3 $$ may be constructed ``one column at a time'' by applications of successive specializations of the Macdonald operator $D_n(X;q,t)$. More precisely they set $$ {\bf B_n}^{(r)}\ses D(-1/qt^{n-r-1};q,t)\, \ue_r $$ and note that if $\nu$ is any partition with no more than $r\leq n$ parts, and $\mu$ is obtained by adding a column of length $r$ to $\nu$, then $$ J_\mu[X;q,t]\ses {1 \over \prod_{i=r+1}^n(1-q^{-1}t^{r-i+1}) }\ess {\bf B}_n^{(r)}\, J_\nu[X;q,t\, ]\ess . \eqno 3.4 $$ Recalling that we have set $$ \TH_\mu[X;q,t]\ses t^{n(\mu)}J_\mu[{\textstyle{X\over 1-1/t}};q,1/t\,]\ess , \eqno 3.5 $$ we can easily see that when $\TH_\mu[X;q,t]$ is constructed by combining 3.5 with the recurrence in 3.4, the $\TK_{\la\mu}(q,t)\, 's$ will necessarily come out as polynomials in ${\bf Z}[q,t]$ divided by factors of the form $$ 1-t^r\ess\scs\ess\ess\ess q-t^r\ess\scs\ess\ess\ess t^rq^s\ess . $$ Now another consequence of the Macdonald duality theorem is that we have $$ \TK_{\la\mu}(q,t)\ses \TK_{\la\mu'}(t,q)\ess . \eqno 3.6 $$ This shows that $\TK_{\la\mu}(q,t)$ itself may also be given an expression consisting of a polynomial in ${\bf Z}[q,t]$ divided by factors of the form $$ 1-q^r\ess\scs\ess\ess\ess t-q^r\ess\scs\ess\ess\ess t^rq^s\ess . $$ Comparing these two sets of factors we see that each of these two different expressions for $\TK_{\la\mu}(q,t)$ must in the end simplify to the point that the only remaining factors are of the form $$ 1-t\ess\scs\ess\ess\ess 1-q\ess\scs\ess\ess\ess t-q\ess\scs\ess\ess\ess t^rq^s\ess . $$ Now specializations at $t=1$ or at $q=1$ have been given by Macdonald (see ex.~7, p.~364 of [15]) yielding, for instance, that $$ \TH_\mu[X;q,1]\ses \prod_{i=1}^{l(\mu)}h_{\mu_i}\big[{\textstyle {X\over 1-q}\big](1-q)(1-q^2)\cdots (1-q^{\mu_i})} $$ from which we can easily derive that $\TK_{\la\mu}(q,1)\in {\bf Z}[q,t]$. This excludes at once both $1-q$ and $1-t$ as possible denominator factors. Similarly, it is also shown in [15] ((8.12) p.~354) that $K_{\la\mu}(0,t)$ is none other than the ``Kostka-Foulkes'' coefficient. This, together with 3.6, eliminates at once both factors $t^r$ and $q^s\, ,$ leaving only powers of $$ t-q \eqno 3.7 $$ as possible denominators! In conclusion, to complete the proof of 3.2 a) and b), we only need a result expressing $\TK_{\la\mu}(q,t)$ as a rational function with denominator factors coprime with $t-q$. Our formula I.11 provides precisely such an expression. In fact, the two sources of denominators in I.11 are the application of $\nabla^{-1}$ and the plethystic substitution of $1/(1-t)(1-q)$. However, it is easily seen from the definition in I.1 that the latter only introduces denominator factors of the form $(1-t^i)(1-q^i)$, and this is sufficient for our purposes here. As for $\nabla^{-1}$, we can use the identities we have already collected in this section and derive that the only denominator factors it can possibly introduce are powers of $t$, $q$ and $M$. To see this, let us assume, by induction, that for all $g\in \LA_{{\bf Z}[q,t]}$, which are homogeneous of degree $k-1$, we have $\nabla^{-1}\, g\in \LA_{{\bf Z}[q,t,1/q,1/t,1/M]}$. By Theorem 2.1, we can complete the induction by proving that we also have $\nabla^{-1}\, f\in \LA_{{\bf Z}[q,t,1/q,1/t,1/M]}$ when $f=D_1 g\ess $ or $f=\ue_1 g\ssp $, with $g=S[X;q,t]/M^{k-1}$ and $S$ homogeneous of degree $k-1$. Now in the first case the identity in 2.11 gives $$ \nabla^{-1}\, f\ses \nabla^{-1}\,D_1\, S[X;q,t]/ M^{k-1}\ses -\ue_1\, \nabla^{-1} S[X;q,t]/ M^{k-1}\ess , $$ and in the second case, we can apply 2.21 and derive that $$ \nabla^{-1}\, f\ses \nabla^{-1}\,\ue_1\, S[X;q,t]/ M^{k-1}\ses D_1^*\, \nabla^{-1} S[X;q,t]/ M^{k-1}\ess . $$ Since $\ue_1$ introduces no denominators and $D_1^*$ (see 1.8 b), at the worst, introduces powers of $qt$ in the denominator, the induction hypothesis yields that in both cases we must have $f\in \LA_{{\bf Z}[q,t,1/q,1/t,1/M]}$ as desired. This completes our proof that $\TK_{\la\mu}(q,t)\in {\bf Z}[q,t]$. \sas \heading{\bol Remark 3.1} We should mention that a more refined argument (see [2]) proves that $\nabla$ itself is ``integral`` and $\nabla^{-1}$ is ``Laurent''. More precisely we have $$ \nabla \LA_{{\bf Z}[q,t]}\con \LA_{{\bf Z}[q,t]} \ess\ess\ess \hbox{and}\ess\ess\ess\ess \nabla^{-1}\LA_{{\bf Z}[q,t]}\con \LA_{{\bf Z}[q,t,{1\over t},{1\over q}]}\ess , $$ and this is best possible. \sas The next application is our derivation of the symmetric function results of Sahi [16] and Knop~[11], [12]. Since these two works are very closely related we shall deal only with Sahi's case here. \sas The results we are concerned with here may be stated as follows: \sas \heading{\bol Theorem 3.1 \rm (Sahi)} {\ita For any $\mu \part n$ there is a unique polynomial $R_\mu[X;q,t\, ]\in \LA_{{\bf Z}[q,t]}$ with the vanishing properties $$ R_\mu\big[\,{\textstyle {\sum_{i=1}^n}t^{-n+i}q^{-\nu_i}};q,t\, ]\ses 0 \ess\ess\ess\ess \hbox{for all $|\nu|\leq |\mu|\ssp \&\ssp \nu\neq \mu $ } \eqno 3.8 $$ and the normalization $$ R_\mu\big[\,{\textstyle {\sum_{i=1}^n}t^{-n+i}q^{-\mu_i}};q,t\, ]\ses 1\ssp . \eqno 3.9 $$ This polynomial can also be characterized, up to a scalar factor, by the difference equation $$ {\tilde D}_1\, R_\mu[X;q,t\, ]\ses \Big( {\textstyle{{\, \, \, 1-t^n\over 1-t}\, -\, \sum_{i=1}^n}t^{n-i}q^{\mu_i}}\Big) R_\mu[X;q,t\, ]\, , \eqno 3.10 $$ where $\tilde D_1$ is the non-homogeneous difference operator $$ {\tilde D}_1\ses \sum_{i=1}^n \Big( {\prod_{\multi{j=1 \cr j\neq i\cr }}^n {\textstyle tx_i-x_j\over x_i-x_j}} \Big) \big( 1- {\textstyle {1\over x_i}}\big)\big( 1- T_q^{(i)}\big) \eqno 3.11 $$ and $T_q^{(i)}$ is the operator that changes $x_i$ into $qx_i$ in a polynomial in% \break $x_1,x_2,\ldots,x_n$. } \sas Our results not only explicitly identify $R_\mu[X;q,t\, ]$ as an image of the Macdonald polynomial $P_\mu$, but also determine the values taken by the left hand side of 3.8 for all the other choices of $\mu$. Moreover we can show that the difference equation in 3.10 is itself the appropriately ``shifted'' image of the Macdonald difference operator. \sa To be precise we have: \heading{\bol Theorem 3.2} {\ita The polynomial $R_\mu[X;q,t\, ]$ may be obtained by deforming the polynomial defined in I.22 according to the following equation: $$ R_\mu[X;q,t\, ]\ses \delta_\mu\big[ t^n(1-1/t)X-t^n; q^{-1},t\,\big]\ess . \eqno 3.12 $$ In particular we must also have $$ R_\mu\Big[{\textstyle{\sum_{i=1}^n t^{-n+i}q^{-\nu_i}}} \,\, ;q \,,t\Big]\ses \cases { \TH_{\nu/\mu}[1;q^{-1},t\, ] & if $\mu\con \nu\,$,\cr\cr 0 & otherwise. \cr } \eqno 3.13 $$ Moreover, the difference equation in 3.10 may be obtained by applying the corresponding deformation to the difference equation $$ D_0^*\, \delta_\mu[X ;q,t\, ]\sms {\textstyle{M\over tq}}\del_1\,\,\delta_\mu[X ;q,t\, ] \ses -D_\mu(1/q,1/t)\, \delta_\mu[X ;q,t\, ] \eqno 3.14 $$ which characterizes our polynomial $\delta_\mu[X;q,t\, ]$. } \sas \heading{\bol Proof} For convenience, let us set as Sahi does $$ \overline \mu(q,t)\ses \sum_{i=1}^n t^{-n+i}q^{-\mu_i}\ess . $$ The definition in I.3 can then be rewritten as $$ MB_\mu(q,t)\ses 1-t^n-t^{n-1}(1-t)\, {\overline \mu}(q^{-1},t)\ess . $$ Thus from I.4 we obtain that $$ D_\nu(q^{-1},t)\ses t^n(1-1/t)\,\, {\overline \nu}(q ,t)- t^n\ess . \eqno 3.15 $$ Making the replacements $q\rightarrow q^{-1}$ and $\la\RA\nu$ in I.24, we get $$ \delta_\mu\Big[t^n(1-1/t)\,\, {\overline \nu}(q ,t)- t^n\,\, ;q^{-1},t\Big]\ses \cases { \TH_{\nu/\mu}[1;q^{-1},t\, ] & if $\mu\con \nu\,$,\cr\cr 0 & otherwise. \cr } \eqno 3.16 $$ In particular, we obtain that the polynomial $\delta_\mu\Big[t^n(1-1/t)\,\, X- t^n\,\, ;q^{-1},t\Big]$ satisfies the conditions in 3.8 and 3.9 that characterize $R_\mu[X;q,t\, ]$. This proves the identity in 3.12 and thus 3.13 follows from 3.16 . To prove 3.14 we start by noting that 1.36 b) gives $$ \TAU_\eee^{-1} D_0^*\ses D_0^*\, \TAU_\eee^{-1}\sps D_{-1}^* \TAU_\eee^{-1}\ess . $$ Thus the Macdonald equation $$ D_0^*\, \TH_\mu[X;q,t\, ] \ses -D_\mu({\textstyle{1 \over q},{1\over t}} )\,\TH_\mu[X;q,t\, ] $$ may be converted to $$ D_0^*\, \TAU_\eee^{-1}\, \TH_\mu[X;q,t\, ] \sps D_{-1}^* \TAU_\eee^{-1}\, \, \TH_\mu[X;q,t\, ] \ses -D_\mu({\textstyle{1 \over q},{1\over t}} )\, \TAU_\eee^{-1}\TH_\mu[X;q,t\, ] \ess . $$ Applying $\nabla^{-1}$ to both sides and using the commutativity of $D_0^*$ and $\nabla$, we can write $$\displaylines{ D_0^*\,\nabla^{-1} \TAU_\eee^{-1}\, \TH_\mu [X;q,t\, ]+ \nabla^{-1} D_{-1}^*\nabla \nabla^{-1} \TAU_\eee^{-1}\, \, \TH_\mu[X;q,t\, ] \hskip.5in\cr\hskip2.5in = -D_\mu({\textstyle{1 \over q},{1\over t}})\,\nabla^{-1} \TAU_\eee^{-1}\TH_\mu[X;q,t\, ] \ess . \cr} $$ Thus the definition in I.22 gives $$ D_0^*\delta_\mu [X;q,t\, ]\sps \nabla^{-1} D_{-1}^*\nabla\,\delta_\mu [X;q,t\, ]\ses -D_\mu({\textstyle{1 \over q},{1\over t}})\, \delta_\mu [X;q,t\, ]\ess . \eqno 3.17 $$ On the other hand, 1.24 b) and 1.30 a*) give $$ -D_{-1}^*\ses D_0^*\del_1 -\del_1 D_0^*\ses {\textstyle{M\over qt}}\, \nabla \del_1\nabla^{-1} $$ reducing 3.17 to $$ D_0^*\delta_\mu [X;q,t\, ]\sms {\textstyle{M\over qt}} \del_1\,\delta_\mu [X;q,t\, ]\ses -D_\mu({\textstyle{1 \over q},{1\over t}})\, \delta_\mu [X;q,t\, ] \eqno 3.18 $$ as desired. \sas We are left to show that 3.10 is just another way of writing 3.18. To this end, we recall the definition in 1.8 b), and write 3.18 as $$ \delta_\mu\big[X-{\textstyle{M\over z qt}}\, ;q,t\, ]\OM[zX]\, \big|_{z^0}\sms {\textstyle{M\over qt}}\, \del_1\, \delta_\mu[X;q,t\, ]\ses -D_\mu({\textstyle{1 \over q},{1\over t}}) \delta_\mu[X;q,t\, ]\ess . $$ Making the replacements $q\RA1/q\scs X\RA t^n(1-1/t)X-t^n$ gives $$ \eqalign{ \delta_\mu\big[t^n&(1-1/t)X-t^n\,-\, {\textstyle{(1-1/t)(1-q)\over z }}\, ;q^{-1},t\, ] \OM[z(t^n(1-1/t)X-t^n)]\, \big|_{z^0}\cr & \sms {\textstyle{(1-1/t)(1-q)}}\, (\del_1\, \delta_\mu)[t^n(1-1/t)X-t^n;q^{-1},t\, ]\cr &\hskip 1.2in =\;-D_\mu({\textstyle{ q},{1\over t}}) \delta_\mu[t^n(1-1/t)X-t^n;q^{-1},t\, ]\ess . \cr } \eqno 3.19 $$ Since $$ (\del_1\, \delta_\mu)[t^n(1-1/t)X-t^n;q^{-1},t ] \,=\, {1\over t^n(1-1/t)}\, \del_1 \Big(\delta_\mu [t^n(1-1/t)X-t^n;q^{-1},t ]\Big) $$ we can use 3.12 and rewrite 3.19 as $$ \eqalign{ R_\mu\big[ X \,-\, &{\textstyle{ 1-q \over t^n z }}\, ;q ,t\, ]\ssp \OM\big[z t^n(1-1/t)X\big](1-t^nz)]\, \big|_{z^0}\cr &\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess \ess\ess\ess\ess\ess\ess\ess\ess \sms {\textstyle{ 1-q \over t^n}}\, \del_1\, R_\mu [ X ;q ,t\, ]\ses \cr &\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess \ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess -D_\mu({\textstyle{ q},{1\over t}})\, R_\mu[X;q ,t\, ]\, . \cr } $$ Since we can make the replacement $zt^n\RA z$ before taking the coefficient of $z^0$, this equation is equivalent to $$ \eqalign{ R_\mu\big[ X \,-\, &{\textstyle{ 1-q\over z }}\, ;q ,t\, ]\ssp \OM\big[z (1-1/t)X\big](1- z)]\, \big|_{z^0}\cr &\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\sms {\textstyle{ 1-q\over t^n}}\, \del_1\, R_\mu [ X ;q ,t\, ]\ses \cr &\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess \ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess -D_\mu({\textstyle{ q},{1\over t}})\, R_\mu[X;q ,t\, ]\, . \cr } \hskip -.5in \eqno 3.20 $$ Simple manipulations yield that $$ -t^n\, D_\mu(q,1/t)\, \ses 1-(1-t)\sum_{i=1}^n\, t^{n-i}q^{\mu_i}\ess . $$ This given, multiplying 3.20 by $\, t^n/(1-t)$, and adding ${t^n\over t-1}R_\mu$ to both sides we finally obtain that $$ \matrix{ {\textstyle{ t^n\over t-1 }}\, R_\mu[X;q,t\, ] \hfill\cr\hskip.5in \;+\; {\textstyle{ t^n\over 1-t }}\, R_\mu\big[ X - {\textstyle{ 1-q\over z }}\, ;q ,t\, \big]\ssp \OM\big[z (1-1/t)X\big](1- z)]\, \big|_{z^0} \hfill \cr\hskip 1in \sms{\textstyle{ 1-q \over 1-t}}\, \del_1\, R_\mu [ X ;q ,t\,] \ses \big({\textstyle{{\,\,\,1-t^n\over 1-t}-\sum_{i=1}^n\, t^{n-i}q^{\mu_i}}}\big)\ess R_\mu[X;q ,t\, ] \cr } \hskip-.4in \eqno 3.21 $$ %$$ %\eqalign{ %{\textstyle{ t^n\over t-1 }}\, R_\mu[X;q,t\, ]+{\textstyle{ t^n\over 1-t }}\, %R_\mu\big[ X - &{\textstyle{ 1-q\over z }}\, ;q ,t\, \big]\ssp \OM\big[z (1-1/t)X\big](1- z)]\, %\big|_{z^0}\cr %&\ess\ess\ess\ess\ess\ess\ess\ess\bigsp %\sms{\textstyle{ 1-q \over 1-t}}\, \del_1\, R_\mu [ X ;q ,t\, %]\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess\ess 3.21\cr &\ess\ess\ess\ess\ess\ess\bigsp\bigsp %\ses\big({\textstyle{{\,\,\,1-t^n\over 1-t}-\sum_{i=1}^n\, t^{n-i}q^{\mu_i}}}\big)\ess R_\mu[X;q ,t\, ] %\cr %} %$$ Now we shall show in Section 4 that the Sahi operator $\tilde D_1$ may also be given the plethystic form $$ {\tilde D_1}P[X]= {\textstyle{ t^n\over t-1}}\,P[X]+{\textstyle{ t^n\over 1-t }}\, P[X-{\textstyle{ 1-q\over z }}\big]\ssp \OM\big[z (1-1/t)X\big](1- z)\, \big|_{z^0}-{\textstyle{ 1-q \over 1-t}}\, \del_1\,P[X] \, . \eqno 3.22 $$ Thus 3.20 reduces to 3.10 as desired, and our proof is complete. \sa Our final application is the Macdonald-Koornwinder reciprocity formula ([15] eq.~(6.6) p.~332). Simple manipulations allow us to state this identity in the following plethystic form. \sas \heading{\bol Theorem 3.3} {\ita For all pairs of partitions $\la\scs \mu$ we have} $$ \TH_\mu[1+u\, D_\la(q,t);q,t ]\, \OM[uB_\mu (q,t)] \,=\, \TH_\la[1+u\, D_\mu(q,t);q,t ]\, \OM[uB_\la (q,t)] \, . \eqno 3.23 $$ {\bol Proof} Since $\OM[XY]$ is the reproducing kernel for the Hall scalar product, we have $$ \TH_\mu[1+u\, D_\la;q,t\, ]\ses \LL \TH_\mu[X+1;q,t\, ]\scs \OM[uXD_\la ]\RR\ess . $$ Thus, expressing $\TH_\mu[X+1;q,t\, ]$ by means of our formula I.20, we obtain the following remarkable sequence of equalities. $$ \eqalign{ \TH_\mu[1+u\, D_\la;q,t\, ]\, \OM[uB_\mu ] \hskip -.5in\cr &\ses \OM[uB_\mu ]\, \Big\langle \PAU_{1/M}\nabla^{-1}\om \OM\big[{\textstyle{XD_\mu\over M}}\big]\scs \OM[uXD_\la ]\Big\rangle \cr &\ses \OM[uB_\mu ]\, \Big\langle \nabla^{-1}\om \OM\big[{\textstyle{XD_\mu\over M}}\big]\scs \TAU_{1/M} \OM[uXD_\la ]\Big\rangle \cr &\ses \OM[uB_\mu ]\,\OM\big[{\textstyle{uD_\la\over M}}\big]\, \Big\langle \nabla^{-1} \OM\big[{\textstyle{XD_\mu\over M}}\big]\scs \OM[uXD_\la ]\Big\rangle \cr &\ses \OM[uB_\mu ]\,\OM[uB_\la ]\,\OM\big[-{\textstyle{u\over M}}\big]\, \Big\langle \nabla^{-1}\om \OM\big[{\textstyle{XD_\mu\over M}}\big]\scs \OM[uXD_\la ]\Big\rangle \cr {(\rm by\ess 1.40)}&\ses \OM[uB_\mu ]\,\OM[uB_\la ]\,\OM\big[-{\textstyle{u\over M}}\big]\, \Big\langle \nabla^{-1}\om \OM\big[{\textstyle{XD_\mu\over M}}\big]\scs \om \OM\big[{\textstyle{XD_\la\over M}}\big] \Big\rangle_* \cr } $$ and this proves 3.23 since the last expression is symmetric in $\mu$ and $\la$ by virtue of the $*$-self-adjointness of the operator $\nabla$. \sas As a corollary we immediately obtain our version of the Macdonald specialization ([15] (6.17) p.~338). \sas \heading{\bol Theorem 3.4} $$ \TH_\mu[1-u;q,t\, ]\ses \prod_{s\in\mu}\big(1-u \, t^{l_\mu'(s)}q^{a_\mu'(s)}\big) \eqno 3.24 $$ {\bol Proof} Simply set $\la=\emptyset$ in 3.23. \sa \heading{\bol 4. Auxiliary identities} We shall begin by converting some of the basic difference operators to plethystic form. \sas \heading{\bol Theorem 4.1} {\ita For any $P\in \LA$ set $$ \BH_mP]X]\ses P\big[X-{\textstyle {1\over z}}\big]\OM[zX]\, \big|_{z^m} \ess . \eqno 4.1 $$ Then for $\la=(\la_1\geq \la_2\geq \cdots \geq \la_n\geq 0)$ we have } $$ S_\la [X]\ses \BH_{\la_1}\BH_{\la_2}\cdots \BH_{\la_n}\cdot 1 \ess . \eqno 4.2 $$ {\bol Proof} The bideterminatal formula for Schur functions $(\dag)$\FOOTNOTE{}{$(\dag)$\ess (3.1) p.~40 of [15]} may be written in the form $$ S_{\la_1,\ldots ,\la_n}[X_n]\ses \sum_{\sig\in S_{[1,n]}}\sig \Big( {x_{1}^{n-1+\la_1}x_{2}^{n-2+\la_2}\cdots x_{n}^{n-n+\la_n} \over \prod_{1\leq i