Ed Allen - Bitableaux bases for some Garsia-Haiman modules and other related modules

Next: Jean-Christophe Aval - To Up: Algebraic Combinatorics, Group Representations Previous: Algebraic Combinatorics, Group Representations

Ed Allen - Bitableaux bases for some Garsia-Haiman modules and other related modules

ED ALLEN, Wake Forest University, Reynolda Station, Winston-Salem, North Carolina 27109, USA

Bitableaux bases for some Garsia-Haiman modules and other related modules

Let ${\cal A}$ be the alphabet

$\begin{displaymath}{\cal A}=\{\dots, (0,3), (0,2), (0,1), (0,0), (1,0), (2,0), (3,0), \dots \}. \end{displaymath}$

Let C[X,Y,Z,W] be the polynomial ring in the variables $X=\{x_1, x_2, \dots, x_n\}$ , $Y=\{y_1, y_2, \dots, y_n\}$ , $Z=\{z_1, z_2, \dots, z_n\}$ and $W=\{w_1, w_2, \dots, w_n\}$ . Given a subset $S=\{(a_1,b_1), (a_2,b_2), \dots, (a_n,b_n)\}$ of the alphabet ${\cal A}$ , we define M_S to be the $n \times n$ matrix

$\begin{displaymath}M_S = (x_i^{a_j} y_i^{b_j})_{1\le i,j\le n} \end{displaymath}$

and $\Delta_S(X,Y)$ to be the determinant of M_S. Let $\partial_{x_i}$ denote the partial differential operator with respect to x_i. With $P(X,Y)\in C[X,Y]$ , we will set $P(\partial_x,\partial_Y) = P(\partial_{x_1}, \partial_{x_2},\dots,\partial_{x_n},\partial_{y_1}, \partial_{y_2},\dots, \partial_{y_n})$ . Setting ${\cal I}_S$ to be the ideal

$\begin{displaymath}{\cal I}_S=\{P(X,Y)\in C[X,Y]:P(\partial_X,\partial_Y)\Delta_S(X,Y) =0\}, \end{displaymath}$

we define R_S(X,Y) to be the polynomial quotient ring

$\begin{displaymath}R_S(X,Y)= C[X,Y]/{\cal I}_S. \end{displaymath}$

The rings R_S(X,Y) are called the Garsia-Haiman modules.

The action of $\sigma\in S_n$ on $P(X,Y,Z,W) \in C[X,Y,Z,W]$ is defined by setting
$\begin{align*}\sigma P(x_1, x_2, \dots, &{}x_n, y_1, \dots, y_n , z_1, \dots, z_... ...\sigma_1}, \dots, z_{\sigma_n}, w_{\sigma_1}, \dots, w_{\sigma_n}). \end{align*}$
Set
$\begin{gather*}R^+[X,Y,Z,W]=\{P(X,Y,Z,W)\in C[X,Y,Z,W]:\sigma P=P\ \forall \sigm... ...al{y}, \partial{z}, \partial{w}) \Delta_S(X,Y)\Delta_T(Z,W) = 0\} \end{gather*}$
and

R_S⁺ (X,Y,Z,W)= C[X,Y,Z,W]/I_S,T.

Analogously, with $\textrm{sgn}(\sigma)$ denoting the sign of the permutation $\sigma$ , let
$\begin{gather*}R^-(X,Y,Z,W)=\{P(X,Y,Z,W)\in C[X,Y,Z,W]:\sigma P = \textrm{sgn}(... ...tial{y},\partial{z},\partial{w}) \Delta_S(X,Y)\Delta_T(Z,W) = 0\} \end{gather*}$
and

R^-_S,T (X,Y,Z,W)= C[X,Y,Z,W]/J_S,T.

We construct bases for R_S(X,Y), R⁺_S,T(X,Y,Z,W) and R^-_S,T(X,Y,Z,W) (for certain generalclasses of S and T that are called dense) that are indexed by pairs of standard tableaux and sequences $\psi_S$ and $\psi_T$ .

Next: Jean-Christophe Aval - To Up: Algebraic Combinatorics, Group Representations Previous: Algebraic Combinatorics, Group Representations