%% Ensure that fpsac.cls is in the same directory %% as this document \documentclass{amsart} \evensidemargin 0in \oddsidemargin \evensidemargin \textwidth 6.5in \tolerance 5000 %% Theorem definitions \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{claim}{Claim}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{cor}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{case}{Case} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \numberwithin{figure}{section} \usepackage{color} \usepackage{amsmath, latexsym, amssymb} \usepackage{graphicx} \usepackage{amsfonts} \usepackage{subfigure} \usepackage{amsthm} \usepackage{amsmath} \usepackage{graphics} \usepackage{amssymb} \usepackage{latexsym} \usepackage{amscd} \usepackage{color} \usepackage{graphicx} \usepackage{subfigure} \usepackage{pict2e} \usepackage[all]{xy} \usepackage{setspace} \DeclareMathOperator{\diagram}{dg} \newcommand{\skydg}{\diagram'} \newcommand{\augdg}{\widehat{\diagram}} \DeclareMathOperator{\arm}{\text{arm}} \DeclareMathOperator{\leg}{\text{leg}} \DeclareMathOperator{\Des}{Des} \DeclareMathOperator{\Inv}{Inv} \DeclareMathOperator{\Coinv}{Coinv} \DeclareMathOperator{\maj}{maj} \DeclareMathOperator{\comaj}{comaj} \DeclareMathOperator{\inv}{inv} \DeclareMathOperator{\coinv}{coinv} \DeclareMathOperator{\cocharge}{cc} \DeclareMathOperator{\cword}{cw} \DeclareMathOperator{\ainv}{ainv} \DeclareMathOperator{\amaj}{amaj} \DeclareMathOperator{\sinv}{sinv} \DeclareMathOperator{\smaj}{smaj} \DeclareMathOperator{\SYT}{SYT} \DeclareMathOperator{\SSYT}{SSYT} \DeclareMathOperator{\wt}{wt} \DeclareMathOperator{\Yam}{Yam} \DeclareMathOperator{\ch}{ch} \DeclareMathOperator{\col}{col} \DeclareMathOperator{\key}{key} \DeclareMathOperator{\colform}{colform} \DeclareMathOperator{\st}{\mathfrak{U}(\omega,\lambda)} %%% tableau macros %%% \newlength\cellsize \setlength\cellsize{15\unitlength} \savebox2{% \begin{picture}(15,15) \put(0,0){\line(1,0){15}} \put(0,0){\line(0,1){15}} \put(15,0){\line(0,1){15}} \put(0,15){\line(1,0){15}} \end{picture}} \newcommand\cellify[1]{\def\thearg{#1}\def\nothing{}% \ifx\thearg\nothing \vrule width0pt height\cellsize depth0pt\else \hbox to 0pt{\usebox2\hss}\fi% \vbox to 15\unitlength{ \vss \hbox to 15\unitlength{\hss$#1$\hss} \vss}} \newcommand\tableau[1]{\vtop{\let\\=\cr \setlength\baselineskip{-16000pt} \setlength\lineskiplimit{16000pt} \setlength\lineskip{0pt} \halign{&\cellify{##}\cr#1\crcr}}} \savebox3{% \begin{picture}(15,15) \put(0,0){\line(1,0){15}} \put(0,0){\line(0,1){15}} \put(15,0){\line(0,1){15}} \put(0,15){\line(1,0){15}} \end{picture}} \newcommand\expath[1]{% \hbox to 0pt{\usebox3\hss}% \vbox to 15\unitlength{ \vss \hbox to 15\unitlength{\hss$#1$\hss} \vss}} %%%%% end tableaux macros %%%%%%%%%%%%%%%%%%%%%%%%%%%% \author[S. Mason]{S. Mason} \address{Department of Mathematics, Davidson College} \email{samason@davidson.edu} \urladdr{http://www.davidson.edu/math/mason} \title[Demazure atoms]{Comment on `A Decomposition of Schur Functions and an Analogue of the Robinson-Schensted-Knuth Algorithm'} \newcommand{\C}{\mathbb{C}} \begin{document} \maketitle The purpose of this comment is to clarify the connections between Demazure characters and the objects studied in this work. The nonsymmetric Macdonald polynomials introduced by Macdonald \cite{Mac} and studied by Cherednick \cite{Che} are denoted by $E_{\alpha} (X; q,t)$, where $\alpha$ is a weak composition and $X=(x_1,x_2, \hdots)$. The Demazure characters introduced by Demazure in \cite{Dem} and studied by Ion~\cite{Ion}, Joseph~\cite{Jos}, and Sanderson~\cite{San} are the specializations $E_{\alpha}(X;0,0)$. Marshall \cite{Mar} works with a variation of the above nonsymmetric polynomials obtained by reversing the indexing composition, reversing the variables, and replacing $q$ and $t$ by $q^{-1}$ and $t^{-1}$ respectively. These nonsymmetric polynomials, denoted $\hat{E}_{\alpha}(X;q,t)$, can therefore be written as $\hat{E}_{\alpha}(x_1,x_2,\hdots;q,t)=E_{{\rm reverse}(\alpha)}(\hdots, x_2, x_1; q^{-1},t^{-1})$. It is these polynomials that we specialize to obtain the polynomials explored in this paper. In fact, the specializations of the $\hat{E}_{\alpha}(X;q,t)$ to $q=t=0$ are equivalent to the second family of Demazure characters, often called ``standard bases" or ``Demazure atoms", introduced by Lascoux and Sch\"utzenberger in \cite{LS90} and studied by Lascoux in~\cite{Las}. Please see \cite{M06} for a combinatorial proof of this equivalence. We provide the following short table for the partition $\lambda=(2,1,0)$ to illustrate the distinction between $E_{\alpha}(X;0,0)$ and $\hat{E}_{\alpha}(X;0,0)$. \begin{center} \begin{tabular}{|c|c|c|} \hline Composition $\alpha$ & $E_{\alpha}(X;0,0)$ & $\hat{E}_{\alpha}(X;0,0)$ \\ \hline $(2,1,0)$ & $x_1^2x_2$ & $x_1^2 x_2$\\ \hline $(2,0,1)$ & $x_1^2x_2 + x_1^2x_3$ & $x_1^2 x_3$\\ \hline $(1,2,0)$ & $x_1^2x_2+x_1x_2^2$ & $x_1 x_2^2$\\ \hline $(1,0,2)$ & $x_1^2x_2 + x_1x_2^2 + x_1^2x_3 + x_1 x_2 x_3 + x_1 x_3^2$ & $x_1 x_2 x_3 + x_1 x_3^2$\\ \hline $(0,2,1)$ & $x_1^2x_2 + x_1x_2^2 + x_1^2 x_3 + x_1 x_2 x_3 + x_2^2 x_3$ & $x_1 x_2 x_3 + x_2^2 x_3$\\ \hline $(0,1,2)$ & $x_1^2x_2 + x_1x_2^2 + x_1^2 x_3 + 2x_1 x_2 x_3 + x_2^2 x_3 + x_1x_3^2 + x_2 x_3^2$ & $x_2 x_3^2$\\ \hline \end{tabular} \end{center} \bibliographystyle{amsalpha} \begin{thebibliography}{A} \bibitem{Che} Cherednik, I,. Nonsymmetric Macdonald polynomials, {\it Math. Res. Notices}, {\bf 10} (1995), pp. 483--515. \bibitem{Dem} Demazure, M., D\'{e}singularisation des vari\'{e}t\'{e}s de Schubert, {\it Ann. E. N. S.}, 6 (1974), 163--172. \bibitem{Ion} Ion, Bogdan, Nonsymmetric {M}acdonald polynomials and {D}emazure characters, {\it Duke Mathematical Journal}, 116 (2003), 299--318 \bibitem{Jos} Joseph, A., On the {D}emazure character formula, {\it Ann. Sci. \'Ecole Norm. Sup.} (4), 18:3 (1985), 389--419. \bibitem{LS90} Lascoux, A., and Sch{\"u}tzenberger, M.-P., Keys and Standard Bases, {\it Invariant Theory and Tableaux, IMA Volumes in Math and its Applications} (D. Stanton, ED.), Southend on Sea, UK, 19 (1990), 125--144. \bibitem{Las} A. Lascoux, Double Crystal graphs, Studies in Memory of Issai Schur, {\it Progress In Math.} 210, Birkha\"user (2003) 95--114. \bibitem{Mac} Macdonald, I. G., {\it Affine Hecke algebras and orthogonal polynomials}, Ast\'{e}risque {\bf 237} (1996), pp.189--207, S\'{e}minaire Bourbaki 1994/95, Exp. no. 797. \bibitem{Mar} Dan Marshall, Symmetric and nonsymmetric {M}acdonald polynomials. {O}n combinatorics and statistical mechanics, {\it Ann. Comb.} \textbf{3} (1999), no.~2-4, 385--415. \bibitem{M06} Mason, S., An explicit construction of type A Demazure Atoms, to appear in {\it J. Algebraic Combinatorics}. \bibitem{San} Sanderson, Y., On the Connection between Macdonald polynomials and Demazure characters, {\it J. Algebraic Combin.} 11 (2000), no.3, 269-275. \end{thebibliography} \end{document} %----------------------------------------------------------------------- % End of article.tex %-----------------------------------------------------------------------