\documentclass[12pt,reqno]{amsart} %\usepackage{pdfsync} \usepackage{amssymb} \usepackage{graphicx} \usepackage{epsf} \usepackage{amsmath} \usepackage[latin1]{inputenc} \usepackage{amsfonts} \usepackage{color} %\usepackage[notref,notcite]{showkeys} \headheight=8pt \topmargin=0pt \textheight=624pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \numberwithin{equation}{section} \newtheorem{theo}{Theorem}[section] \newtheorem{prop}[theo]{Proposition} \newtheorem{conj}[theo]{Conjecture} \newtheorem{lemm}[theo]{Lemma} \newtheorem{coro}[theo]{Corollary} \newtheorem{defi}[theo]{Definition} \newtheorem{prob}[theo]{Problem} \newtheorem{rema}[theo]{Remark} \newtheorem{exem}[theo]{Example} \newenvironment{Remark}{\begin{rema}\sl}{\end{rema}} %%%%%% Environnement resume %\newenvironment{resume}{\footnotesize\quotation} %%%% Fin de l'Environnement resume \newcommand{\pref}[1]{(\ref{#1})} \newcommand{\abs}[1]{|#1|} \newcommand{\Span}[1]{\left\langle #1\right\rangle} \newcommand{\twolines}[2]{{\scriptstyle #1 \atop \scriptstyle \vphantom{\sum}#2}} \def\rouge{\textcolor{red}} \def\bleu{\textcolor{blue}} \def\vert{\textcolor{green}} \def\blan{\textcolor{white}} \def\cf{{\it cf. }} \def\ie{{\it i.e. }} \def\gex{\geq_{\rm lex}} \def\lex{\leq_{\rm lex}} \def\gox{\geq_{\rm opp}} \def\lox{\leq_{\rm opp}} \def\gx{>_{\rm lex}} \def\lx{<_{\rm lex}} \def\go{>_{\rm opp}} \def\lo{<_{\rm opp}} \def\fact{!} \def\ti{$\sim$} \def\shu{\amalg} \def\nm{_{n,m}} \def\concat{{\cdot}} \def\C{{\mathbb C}} \def\N{{\mathbb N}} \def\Z{{\mathbb Z}} \def\Q{{\mathbb Q}} \def\I{{\mathcal I}} \def\J{{\mathcal J}} \def\A{{\mathcal A}} \def\L{{\mathcal L}} \def\D{{\mathcal D}} \def\DI{{\mathcal DI}} \def\S{{\mathbb S}} \def\F{{\mathcal F}} \def\G{{\mathcal G}} \def\T{{\mathcal T}} \def\B{{\mathcal B}} \def\P{{\mathcal P}} \def\X{{\mathcal X}} \def\Y{{\mathcal Y}} \def\P{{\mathcal P}} \def\CC{{\mathcal C}} \def\RR{{\mathcal R}} \def\MM{{\mathcal M}} \def\ZZ{{\mathcal Z}} \def\fh{{\mathfrak h}} \def\bp{{\bf p}} \def\bq{{\bf q}} \def\QSym{{\rm QSym}} \def\DQ{{\rm DQ}} \def\QQ{{\rm Q}} \def\NSym{{\rm NSym}} \def\DNSym{{\rm DNSym}} \def\DSSym{{\rm DSSym}} \def\SSym{{\rm SSym}} \def\DQSym{{\rm DQSym}} \def\DSym{{\rm DSym}} \def\Sym{{\rm Sym}} \def\DSH{{\rm DTLH}} \def\DH{{\rm DH}} \def\TL{{\rm TL}} \def\TLH{{\rm TLH}} \def\Ker{{\rm Ker}} \def\sign{{\rm sign}} \def\DI{\J} \def\V{{\mathcal V}} \def\W{{\bf W}} \def\H{{\bf H}} \def\SH{{\bf SH}} \def\M{{\bf M}} \def\R{{\bf R}} \def\BB{{\bf B}} \def\bD{{\bf D}} \def\shuffle{{\,\raise 1pt\hbox{$\scriptscriptstyle\cup{\mskip-4mu}\cup$}\,}} \def\sshuffle{{\,\hbox{$\scriptscriptstyle\cup{\mskip-4mu}\cup$}\,}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \title[Diagonal Temperley--Lieb invariants and Harmonics]{Diagonal Temperley--Lieb Invariants\\ and Harmonics} \author{J.-C.~Aval}\address[Jean-Christophe Aval]{LaBRI\\ Universit\'e Bordeaux 1\\ 351 cours de la Lib\'eration\\ 33405 Talence cedex\\ FRANCE} \email{aval@labri.fr} \urladdr{http://www.labri.fr/\ti aval/} \author{F.~Bergeron} \address[F. Bergeron]{D\'epartement de Math\'ematiques\\ Universit\'e du Qu\'ebec \`a Montr\'eal\\ Montr\'eal, Qu\'ebec, H3C 3P8, CANADA} \email{bergeron.francois@uqam.ca} \author{N.~Bergeron} \address[N. Bergeron]{Department of Mathematics and Statistics\\ York University\\ To\-ron\-to, Ontario M3J 1P3\\ CANADA} \email{bergeron@mathstat.yorku.ca} \urladdr{http://www.math.yorku.ca/bergeron} \date{} \dedicatory{This paper is dedicated to Adriano Garsia with all our admiration for his incredible mathematical enthusiasm.} \thanks{J.C. Aval is supported by EC's IHRP Programme, Network ``ACE'', grant HPRN-CT-2001-00272.} \thanks{F. Bergeron is supported in part by NSERC and FCAR} \thanks{N. Bergeron is supported in part by CRC, NSERC and PREA} \begin{document} \begin{abstract} In the context of the ring $\Q[{\bf x},{\bf y}]$, of polynomials in $2\,n$ variables ${\bf x}=x_1,\ldots,x_n$ and ${\bf y}=y_1,\ldots,y_n$, we introduce the notion of diagonally quasi-symmetric polynomials. These, also called {\em diagonal Temperley--Lieb invariants}, make possible the further introduction of the space of {\em diagonal Temperley--Lieb harmonics} and {\em diagonal Temperley--Lieb coinvariant space}. We present new results and conjectures concerning these spaces, as well as the space obtained as the quotient of the ring of diagonal Temperley--Lieb invariants by the ideal generated by constant term free diagonally symmetric invariants. We also describe how the space of diagonal Temperley--Lieb invariants affords a natural graded Hopf algebra structure, for $n$ going to $\infty$. We finally show how this last space and its graded dual Hopf algebra are related to the well known Hopf algebras of symmetric functions, quasi-symmetric functions and noncommutative symmetric functions. \end{abstract} \maketitle { \parskip=0pt\footnotesize \tableofcontents} \parskip=8pt %First page headline in AmS-LaTeX for S\'eminaire Lotharingien de Combinatoire %--first part \thispagestyle{myheadings} \font\rms=cmr8 \font\its=cmti8 \font\bfs=cmbx8 \markright{\its S\'eminaire Lotharingien de Combinatoire \bfs 54A \rms (2006), Article~B54Aq\hfill} \def\thepage{} % % \pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction}\label{intro} \subsection{General framework} \label{framework} As a warm up for of our discussion, it may be best to recall some facts about the interactions between the algebras \begin{equation}\label{dia1} \Sym_n\,\hookrightarrow\, \QSym_n\,\hookrightarrow \, \Q[ {\bf x}] \,, \end{equation} of {\em symmetric polynomials}, {\em quasi-symmetric polynomials}, and of polynomials in $n$ variables ${\bf x}=x_1,x_2,\ldots,x_n$. Some of these facts involve the natural (and well known) action of the symmetric group $\S_n$ on polynomials, defined on $p({\bf x})$ in $\Q[{\bf x}]$ as $$\sigma\cdot p({\bf x})=p(x_{\sigma(1)},x_{\sigma(2)},\ldots,x_{\sigma(n)}),$$ for $\sigma$ in $\S_n$. Clearly, this action extends to a faithful action of the group algebra $\Q \S_n$ on $\Q[{\bf x}]$. Furthermore, it preserves the multiplication of polynomials, we can thus consider the {\em algebra of symmetric polynomials}: $$\Sym_n\ :=\ \left\{ p({\bf x})\in\Q[{\bf x}]\ |\ \sigma\cdot p({\bf x})=p({\bf x})\right\}, $$ which is often denoted $\Q[{\bf x}]^{\S_n}$ by other authors. We part from this convention to stay in the spirit of rest of the notation used in this paper. A classical result of Newton states that $\Sym_n$ is itself a polynomial algebra. This is often made explicit as follows: $$\Sym_n=\Q[e_1,e_2,\ldots,e_n],$$ where $e_k=e_k({\bf x})$ is the coefficient of $t^k$ in $(1+x_1t)(1+x_2t)\cdots(1+x_nt)$. In \cite{hivert} Hivert has introduced an action of $\Q \S_n$ on polynomials which does not preserve the multiplication, but still leads to very interesting results. Before describing this action, let us first recall some notation. We denote $ {\bf x}_{\bf i}^{\bf a}$, the monomial \begin{equation}\label{monome} {\bf x}_{\bf i}^{\bf a}=x_{i_1}^{a_1}x_{i_2}^{a_2}\cdots x_{i_k}^{a_k}, \end{equation} assuming here that ${\bf a}= (a_1,a_2,\ldots,a_k)$ is a sequence of positive integers, $a_i>0$; and that the elements $i_m$, of the subset ${\bf i}=\{i_1,i_2,\ldots ,i_k\}$ of $[n]:=\{1,\dots,n\}$, have been listed in increasing order ($i_mn$. The set of $M_{\mathfrak a}$, for $\ell({\mathfrak a})\leq n$, forms a linear basis of $\DQSym_n$. This basis makes evident that the algebra $\QSym_n$ (see (\ref{abbq})) appears, up to isomorphism, as a subspaces of $\DQSym_n$ in two obvious ways. First as the linear span of the set $$\{\ M_{({\bf a},0)}\ |\ \ell({\bf a})\leq n\ \},\qquad{\rm second\ as \ that\ of\ }\qquad \{\ M_{(0,{\bf b})}\ |\ \ell({\bf b})\leq n\ \},$$ with ${\bf a}$ and ${\bf b}$ compositions. To distinguish these two occurrences, we respectively denote them $\QSym_n^{\bf x}$ and $\QSym_n^{\bf y}$. As we have already mentioned, these are respectively subalgebras of $\Q[{\bf x}]$ and $\Q[{\bf y}]$. To show that, in turn, $\DQSym_n$ is itself a subalgebra of $\Q[{\bf x},{\bf y}]$, we easily adapt the method of \cite{Hoff}. To this end, we introduce the sum, concatenation and semi-shuffle of bicompositions. For ${\mathfrak a}=(\alpha_1,\ldots,\alpha_k)$ and ${\mathfrak b}=(\beta_1,\ldots,\beta_\ell)$, the {\em sum} of ${\mathfrak a}$ and ${\mathfrak b}$ is the length $k+\ell-1$ bicomposition \begin{equation} {\mathfrak a}+ {\mathfrak b}:=(\alpha_1,\ldots,\alpha_k+\beta_1,\ldots,\beta_\ell), \end{equation} with $\alpha_k+\beta_1$ being the vector sum; and the {\em concatenation} of ${\mathfrak a}$ and ${\mathfrak b}$ is the length $k+\ell$ bicomposition \begin{equation} {\mathfrak a}\concat {\mathfrak b}:=(\alpha_1,\ldots,\alpha_k,\beta_1,\ldots,\beta_\ell) \end{equation} The {\em quasi-shuffle} of bicompositions can then be defined as the set described recursively as follows. \begin{equation} {\mathfrak a} \tilde\shuffle {\mathfrak b}:=\begin{cases} \{\mathfrak{a}\}& \text{if\ } \mathfrak{b}=0, \\ \{\mathfrak{b}\}& \text{if\ } \mathfrak{a}=0,\\ \displaystyle \bigcup_{\mathfrak{a}_1\concat \mathfrak{a}_2=\mathfrak{a}} \ \bigcup_{\mathfrak{c}\in \mathfrak{a}_2\tilde\sshuffle\mathfrak{b}_2} \{ \mathfrak{a}_1\concat \mathfrak{b}_1\concat \mathfrak{c}, (\mathfrak{a}_1+\mathfrak{b}_1)\concat \mathfrak{c} \}& \text{otherwise}. \end{cases} \end{equation} where $\mathfrak{b}=\mathfrak{b}_1\concat \mathfrak{b}_2$ is the unique factorization of $\mathfrak{b}$ such that $\mathfrak{b}_1$ has only one part. For example $$\begin{array}{rcl} {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0 \\ 1\ 3\end{smallmatrix}\right)} \tilde\shuffle {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}0\ 2 \\ 1\ 0\end{smallmatrix}\right)} &=& \Big\{ {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 0\ 2 \\ 1\ 3\ 1\ 0\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 2 \\ 1\ 4\ 0\end{smallmatrix}\right)}, {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 0\ 2 \\ 1\ 1\ 3\ 0\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 2 \\ 2\ 3\ 0\end{smallmatrix}\right)} ,{\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 2 \\ 1\ 1\ 3\end{smallmatrix}\right)} ,{\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 2 \\ 2\ 3\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 2\ 0 \\ 1\ 1\ 0\ 3\end{smallmatrix}\right)} ,\cr &&\quad {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}2\ 2\ 0 \\ 2\ 0\ 3\end{smallmatrix}\right)} ,{\scriptstyle \left(\begin{smallmatrix}\vphantom{g}0\ 2\ 0\ 2 \\ 1\ 1\ 3\ 0\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}0\ 2\ 2 \\ 1\ 1\ 3\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}0\ 2\ 2\ 0 \\ 1\ 1\ 0\ 3\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}0\ 4\ 0 \\ 1\ 1\ 3\end{smallmatrix}\right)} , {\scriptstyle \left(\begin{smallmatrix}\vphantom{g}0\ 2\ 2\ 0 \\ 1\ 0\ 1\ 3\end{smallmatrix}\right)} \Big\}\end{array} $$ Following the argument of \cite{Hoff}, we get \begin{equation}\label{DMmult} M_{\mathfrak a}({\bf x},{\bf y}) M_{\mathfrak b}({\bf x},{\bf y}) =\sum_{{\mathfrak c}\,\in\,{\mathfrak a}\tilde\sshuffle{\mathfrak b}} M_{\mathfrak c}({\bf x},{\bf y})\,. \end{equation} Observe that some of the terms, $M_{\mathfrak c}$, in the resulting sum may vanish when $\ell({\mathfrak c})>n$. One notes that the resulting multiplication rule stabilizes as $n\to\infty$, as $n$ becomes larger then the maximum possible length for ${\mathfrak c}$. Hence we can consider the corresponding inverse limit algebra, $\DQSym$, of {\em diagonally quasi-symmetric series}. As a vector space this algebra is clearly generated by the bimonomial basis elements $$M_{\mathfrak a}=\sum_{{\bf i}\subseteq \N,\ |{\bf i}|=\ell({\mathfrak a})} ({\bf x}{\bf y})_{\bf i}^{\mathfrak a},$$ where ${\mathfrak a}$ runs over all bicompositions. The distinctive feature here is that ${\bf i}$ can be any $k$-subset of $\N$, hence $M_{\mathfrak a}$ is a series rather than a polynomial. The multiplication of the $M_{\mathfrak a}$'s in $\DQSym$ is just as in (\ref{DMmult}), with no vanishing terms. To introduce a graded bialgebra structure on $\DQSym$, we define the comultiplication $$\Delta(M_{\mathfrak{a}}) =\sum_{\mathfrak{a}_1\concat \mathfrak{a}_2=\mathfrak{a}} M_{\mathfrak{a}_1} \otimes M_{\mathfrak{a}_2},$$ and counit $$\epsilon(M_{\mathfrak a})=\begin{cases} 1& \text{if\ } {\mathfrak a}=0, \\ 0 & \text{otherwise}. \end{cases}$$ The fact that we actually get a bigraded Hopf algebra, with respect to bidegree, follows from general principles (see \cite{MM65}). Here the bidegree of $M_{\mathfrak a}$ is naturally set to be the bidegree $d({\mathfrak a})$ of ${\mathfrak a}$ (see (\ref{bidegree})). The restriction to $\DQSym_n$ of the map $\pi$, introduced in (\ref{projection}), clearly factors through $\QSym_n$, since for every bicomposition ${\mathfrak a}=({\bf a},{\bf b})$ we evidently have $$\pi(M_{({\bf a},{\bf b})})=M_{{\bf a}+{\bf b}}.$$ This map preserves multiplication, and when $n\to\infty$ it also preserves comultiplication and counit. Observe that $\DQSym$ can also be considered as a simply graded Hopf algebra by setting $$\DQSym^{(n)}:=\bigoplus_{n=d_1+d_2} \DQSym^{(d_1,d_2)},$$ with $\DQSym^{(d_1,d_2)}$ being the bihomogeneous component of bidegree $(d_1,d_2)$ of\break $\DQSym$. We get the following theorem. \begin{theo} $\DQSym$ is a bigraded Hopf algebra and the map $\pi\colon \DQSym \to \QSym$ is a morphism of graded Hopf algebras.\qed \end{theo} We end this section with the description of another important basis of $\DQSym$. Consider the partial order on bicompositions obtained as the transitive closure of the set of covering relations $$\{ {\mathfrak a}_1+{\mathfrak a}_2 < {\mathfrak a} \ |\ {\mathfrak a}_1\concat {\mathfrak a}_2 = {\mathfrak a}\}\,.$$ We introduce the following diagonal analogue of the usual ``fundamental basis'' for quasi-symmetric functions: \begin{equation}\label{fondamental} F_{\mathfrak b}:=\sum_{{\mathfrak b}\le{\mathfrak a}} M_{\mathfrak a}. \end{equation} For instance, we have $$\begin{array}{rcl} F_{ \left(\begin{smallmatrix}\vphantom{g}2\ 0 \\ 1\ 1\end{smallmatrix}\right)} &=& M_{ \left(\begin{smallmatrix}\vphantom{g}2\ 0 \\ 1\ 1\end{smallmatrix}\right)} + M_{ \left(\begin{smallmatrix}\vphantom{g}1\ 1\ 0 \\ 1\ 0\ 1\end{smallmatrix}\right)} + M_{ \left(\begin{smallmatrix}\vphantom{g}1\ 1\ 0 \\ 0\ 1\ 1\end{smallmatrix}\right)} + M_{ \left(\begin{smallmatrix}\vphantom{g}2\ 0\ 0 \\ 0\ 1\ 1\end{smallmatrix}\right)} + M_{ \left(\begin{smallmatrix}\vphantom{g}0\ 2\ 0 \\ 1\ 0\ 1\end{smallmatrix}\right)} + \cr \cr&&M_{ \left(\begin{smallmatrix}\vphantom{g}1\ 1\ 0\ 0 \\ 0\ 0\ 1\ 1\end{smallmatrix}\right)} + M_{ \left(\begin{smallmatrix}\vphantom{g}1\ 0\ 1\ 0 \\ 0\ 1\ 0\ 1\end{smallmatrix}\right)}+ M_{ \left(\begin{smallmatrix}\vphantom{g}0\ 1\ 1\ 0 \\ 1\ 0\ 0\ 1\end{smallmatrix}\right)} \,. \end{array} $$ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The graded Hopf dual \DNSym, and other Hopf properties} Consider the noncommutative free algebra $$\DNSym:=\Q\langle {\mathfrak h}_\alpha \ |\ \alpha\in \N^2\setminus {0} \rangle,$$ of noncommutative polynomials in the variables ${\mathfrak h}_\alpha$, as a bigraded algebra for which we set the bidegree of the variable $\fh_\alpha$, to be $\alpha=(a,b)$. Here, as usual, we denote $0$ the zero vector in $\N^2$. We further introduce a bigraded coalgebra structure on $\DNSym$, setting as comultiplication $$\Delta(\fh_\alpha):=\sum_{\alpha=\beta+\gamma} \fh_\beta\otimes \fh_\gamma,$$ where $\beta+\gamma$ is understood to be the usual vector sum. By convention we let $\fh_{0}:=1$ be the unit of $\DNSym$. The counit is simply the linear form such that $$\epsilon(\fh_\alpha):=\begin{cases} 1 & \text{if\ }\alpha =0, \\ 0 & \text{otherwise}. \end{cases}$$ It is then straightforward to check that $\DNSym$ is in fact a bigraded Hopf algebra. This bigrading corresponds to the decomposition $$\DNSym=\bigoplus_{(d_1,d_2)} \DNSym_n^{(d_1,d_2)}$$ in bihomogeneous components $\DNSym_n^{(d_1,d_2)}$. Each of these bihomogeneous components affords a linear basis $$ \Big\{ \fh_{\mathfrak a} \ | \ {\mathfrak a}=(\alpha_1,\cdots,\alpha_k),\quad \alpha_i=(a_i,b_i), \ {\rm and}\ d({\mathfrak a})=(d_1,d_2)\Big\}.$$ \begin{theo} $\DNSym$ is isomorphic to the bigraded Hopf dual algebra $\DQSym^*$. \end{theo} \proof As a vector space, we clearly must have $$\DQSym^* =\bigoplus_{(d_1,d_2)} {\DQSym_n^{(d_1,d_2)}}^*.$$ We consider the pairing $$\langle \fh_{\mathfrak a},M_{\mathfrak b}\rangle:=\begin{cases} 1 & \text{if\ }{\mathfrak a}={\mathfrak b}, \\ 0 & \text{otherwise}, \end{cases}$$ which gives a graded vector space isomorphism between $\DNSym$ and $\DQSym^*$. To complete our proof, we need to show that the multiplicative and comultiplicative structure is preserved. Let $c_{{\mathfrak a},{\mathfrak b}}^{\mathfrak c}$ be the multiplicative structure constants of $\DQSym$: $$M_{\mathfrak a} M_{\mathfrak b} =\sum_{{\mathfrak c}} c_{{\mathfrak a},{\mathfrak b}}^{\mathfrak c} M_{\mathfrak c}.$$ >From (\ref{DMmult}), we have $$c_{{\mathfrak a},{\mathfrak b}}^{\mathfrak c} = \left\{ \begin{array}{ll} 1 &\mbox{ if } {\mathfrak c} \in {\mathfrak a}\tilde\sshuffle {\mathfrak b} , \medskip \\ 0& \mbox{ otherwise.}\end{array} \right. $$ On the other hand, in $\DNSym$, we have $$\Delta(\fh_{\mathfrak c})=\sum_{{\mathfrak c} \in {\mathfrak a}\tilde\sshuffle {\mathfrak b} } \fh_{\mathfrak a}\otimes\fh_{\mathfrak b}.$$ This implies that $\langle \fh_{\mathfrak c}, M_{\mathfrak a} M_{\mathfrak b}\rangle = \langle\Delta( \fh_{\mathfrak c}), M_{\mathfrak a}\otimes M_{\mathfrak b}\rangle$. In the same way one can show that $\langle \fh_{\mathfrak a} \fh_{\mathfrak b}, M_{\mathfrak c} \rangle = \langle \fh_{\mathfrak a}\otimes \fh_{\mathfrak b}, \Delta(M_{\mathfrak c}) \rangle$, thus ending the proof. \qed \medskip A consequence of this is that $\DQSym$ is cofree, since $\DNSym$ is free. We can also show that $\DQSym$ is free. This is a consequence of a more general fact about quasi-shuffle Hopf algebra (see \cite{Hoff}). To do this, we consider bicompositions as words on the alphabet of biletters $\binom a b$ ordered lexicographically. As is usual in the context of words, we say that a bicomposition ${\mathfrak a}$ is a {\sl Lyndon} bicomposition, if it is strictly smaller than all of the non-trivial circular rearrangement of its letters. The following theorem is a special case of a theorem of \cite{Hoff}: \begin{prop} [Hoffman] \label{hoffman} $\DQSym=\Q[M_{\mathfrak a} \ | \ {\mathfrak a} \hbox{ is Lyndon}]$, and in particular it is free. \end{prop} \begin{proof}[\bf Remark 2.] In \cite{Poirier}, S. Poirier constructed the graded dual to the Mantaci--Reu\-te\-nau\-er Hopf algebra. Part of his goals was to produce analogs of the Hopf algebras $\QSym$ and $\NSym$ for type $B$. For this, Poirier introduced a larger algebra containing the desired Hopf algebras. In fact, this larger algebra is isomorphic to our $\DQSym$. More generalizations and properties of these Hopf algebras are currently being studied (see \cite{Hohlweg, NovThi}). Moreover, in an upcoming work, S. Kapetanovich \cite{Kape} shows that $\DQSym$ is a terminal object in the category of bigraded Hopf algebras equipped with a multiplicative character. She also computes explicitly its even and odd Hopf subalgebras. We should also note that $\DQSym$ coincides as a vector space with the space of $2\times k$ matrix quasi-symmetric functions of \cite{HivThib}. However their multiplicative structure are very different. In particular the $2\times k$ matrices of \cite{HivThib} are not closed under their multiplication rule. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Conjectures} \subsection{The quotient $\Q[{\bf x},{\bf y}]/\langle \DQSym_n^+\rangle$.}We are now interested in the study of the ideal generated by diagonally quasi-symmetric polynomials without constant term, \ie $$\DI_n=\langle M_{\mathfrak a}({\bf x},{\bf y})\rangle_{{\mathfrak a}\neq\emptyset}.$$ This ideal is clearly bihomogeneous. Next, in the spirit of our discussion in section \ref{intro}, we consider the quotient space $$\DQ_n:=\Q[{\bf x},{\bf y}]/\DI_n,$$ as well as the space of {\em diagonal $TL_n$-harmonic polynomials} as $$\DSH_n:=\DI_n^\perp$$ with respect to the scalar product of \pref{scalXY}. These two spaces are isomorphic as bigraded modules. Observe that \begin{equation}\label{tlhxy} \TLH_n\simeq \DSH_n\cap \Q[{\bf x}], \qquad {\rm and}\qquad \TLH_n\simeq \DSH_n\cap\Q[{\bf y}]. \end{equation} Recall that the {\em bigraded Hilbert series} of a bigraded vector subspace $\V$ is by definition $$H_\V(q,t)=\sum_{h,k}q^ht^k\dim(\V_{h,k})$$ where $\V_{h,k}$'s are the components appearing in the decomposition $$\V=\bigoplus_{h,k} \V_{h,k},$$ corresponding to the bigraded structure of $\V$. The Hilbert series considered here are in fact polynomials. To emphasize the symmetries of these polynomials, we use a Cartesian coordinates entries matrix notation for them. Thus, for a polynomial $p(q,t)$ we define the corresponding {\em Hilbert matrix}, $M$, to have entries $$M(i,j)={\rm coefficient\ of\ } t^{k-i+1 }q^{j-1}\ {\rm in\ } p(q,t),$$ with $k$ equal to the degree in $t$ of $p(q,t)$. Thus we have the correspondence $$\begin{array}{l} 3\,t^2+\\ 2\,t+3\,tq+\\ 1+3\,q+4\,q^2+q^3 \end{array}\quad \leftrightarrow \qquad \begin{pmatrix} 3&&&\\ 2&3&&\\ 1&3&4&1\\ \end{pmatrix}$$ Here, we omit $0$ entries. \begin{conj}\label{conjmat} The Hilbert matrix, of the space $\DQ_n$, is the $n\times n$ lower triangular matrix, $M_n$, with all entries in the main diagonal equal to the $(n-1)^{\rm th}$ Catalan number: $$M_n(i,i)=\frac1n{\binom {2(n-1)} {(n-1)}},$$ and all other non zero entries recursively obtained as: \begin{equation}\label{conj_rec} M_n(i,j)=\underset{{j'\leq j}}{\sum_{i'\geq i}}M_{n-1}(i',j'), \end{equation} when $i>j$. \end{conj} \noindent For instance, the Hilbert matrices $M_n$, for $1\le n\le5$, are \begin{eqnarray*} &&M_1=\begin{pmatrix} 1 \end{pmatrix}, \qquad M_2=\begin{pmatrix} 1&\\ 1&1 \end{pmatrix}, \qquad M_3=\begin{pmatrix} 2&&\\ 2&2&\\ 1&2&2 \end{pmatrix}, \\ &&M_4=\begin{pmatrix} 5&&&\\ 5&5&&\\ 3&7&5&\\ 1&3&5&5 \end{pmatrix}, \quad M_5=\begin{pmatrix} 14&&&&\\ 14&14&&&\\ 9&24&14&&\\ 4&14&24&14&\\ 1&4&9&14&14 \end{pmatrix}. \end{eqnarray*} Actually, the values in the first column (and last line) of $M_n$ are known to satisfy recurrence (\ref{conj_rec}). In view of (\ref{tlhxy}) and the isomorphism of graded spaces $\QQ_n\simeq \TLH_n$, this follows immediately from our result of \cite{ABBQ}, giving the following explicit formula for the Hilbert series of $\QQ_n$: \begin{equation} H_{\QQ_n}(q)=\sum_{k=0}^{n-1}\frac{n-k}{n+k}{\binom {n+k} k} q^k\,. \end{equation} In \cite{ABBQ}, we have in fact constructed a nice explicit linear basis of monomials for $\QQ_n$. We are going to describe here, for each $n$, a set $\B_n$ of monomials which we conjecture to be a linear basis of $\DQ_n$. As should be expected this set contains two copies of the basis for $\QQ_n$; one in the ${\bf x}$-variables, and one in the ${\bf y}$-variables. The degrees of monomials in $\B_n$ are all less or equal to $n-1$. Moreover, the construction will make clear that the bigraded enumeration of $\B_n$ agrees with the Hilbert matrix of Conjecture \ref{conjmat}. First, whenever $i+j