The type A, root of unity case

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 14 November 2012

The type A, root of unity case

This section describes the sets ${ℱ}^{(t,J)}$ in the case of the root system of Section 5.2 when $q^2=e^{2\pi i/\ell },$ a primitive $\ell \text{th}$ root of unity, $\ell >2\text{.}$

Let $t\in T\text{.}$ Identify $t$ with a sequence

$$t=(t_1,\dots ,t_n)\in {ℂ}^n,\text{where}t\left(X^{{\epsilon }_i}\right)=t_i\text{.}$$

For the purposes of representation theory (see Theorem 3.6) $t$ indexes a central character (see Section 2.3) and so $t$ can safely be replaced by any element of its $W\text{-orbit.}$ In this case $W$ is the symmetric group, $S_n,$ acting by permuting the sequence $t=(t_1,\dots ,t_n)\text{.}$

The cyclic group $⟨q^2⟩$ of order $\ell$ generated by $q^2$ acts on ${ℂ}^{*}\text{.}$ F ix a choice of a set $\left\{\xi \right\}$ of coset representatives of the $⟨q^2⟩$ cosets in ${ℂ}^{*}\text{.}$ Replace $t$ with the sequence obtained by rearranging its entries to group entries in the same $⟨q^2⟩\text{-orbit,}$ so that

$$t=({\xi }_1{t}^{\left(1\right)},\dots ,{\xi }_k{t}^{\left(k\right)}),$$

where ${\xi }_1,\dots ,{\xi }_k$ are distinct representatives of the cosets in ${ℂ}^{*}/⟨q^2⟩$ and each $t^{\left(j\right)}$ is a sequence of the form

$$t^{\left(j\right)}=(q^{2{\gamma }_1},\dots ,q^{2{\gamma }_r}),\text{with}{\gamma }_1,\dots ,{\gamma }_r\in \{0,1,\dots ,\ell -1\}\text{and}{\gamma }_1\le \dots \le {\gamma }_r\text{.}$$

As in Section 5.11 this decomposition of $t$ into groups induces decompositions

$$Z\left(t\right)=\bigcup _{j=1}^k{Z}_{{\xi }_j}\left(t\right)\text{and}P\left(t\right)=\bigcup _{j=1}^k{P}_{{\xi }_j}\left(t\right),$$

and it is sufficient to analyze the case when $t$ consists of only one group, i.e., all the entries of t are in the same $⟨q^2⟩$ coset.

Now assume that

$$t=(q^{2{\gamma }_1},\dots ,q^{2{\gamma }_n}),\text{with}{\gamma }_1\le \dots \le {\gamma }_n,{\gamma }_i\in \{0,\dots ,\ell -1\}\text{.}$$

Consider a page of graph paper with diagonals labeled by $\dots ,0,1,\dots ,\ell -1,0,1,\dots ,\ell -1,0,1,\dots$ from southwest to northeast. For each local region $(t,J),$ $J\subseteq P\left(t\right),$ we will construct an $\ell \text{-periodic}$ configuration of boxes for which the $\ell \text{-periodic}$ standard tableaux defined below will be in bijection with the elements of ${ℱ}^{(t,J)}\text{.}$ For each $1\le i\le n,$ the configuration will have a box numbered $i,$ ${\text{box}}_i,$ on each diagonal which is labeled ${\gamma }_i\text{.}$ There are an infinite number of such diagonals containing a box numbered $i,$ since the diagonals are labeled in an $\ell \text{-periodic}$ fashion, but each strip of consecutive diagonals labeled $0,1,\dots ,\ell -1$ will contain $n$ boxes. The content of a box $b$ (see [Mac1995, I, Section 1, Exercise 3]) is

$$c\left(b\right)=\text{(the diagonal number of the box}b\text{).}$$

Then

$$\begin{array}{ccc}Z\left(t\right)& =& \{{\epsilon }_j-{\epsilon }_i\mid i<j,{\gamma }_i={\gamma }_j\}\\ & =& \{{\epsilon }_j-{\epsilon }_i\mid i<j,{\text{box}}_i\text{and}{\text{box}}_j\text{are in the same diagonal}\}\end{array}$$

and

$$\begin{array}{ccc}P\left(t\right)& =& \{e_j-e_i\mid \begin{array}{c}i<j\text{and}{\gamma }_j={\gamma }_i+1,\text{or}\\ i<j,{\gamma }_j=\ell -1,\text{and}{\gamma }_i=0\end{array}\}\\ & =& \{{\epsilon }_j-{\epsilon }_i\mid i<j\text{and}{\text{box}}_i\text{and}{\text{box}}_j\text{are in adjacent diagonals}\}\text{.}\end{array}$$

We will use $J\subseteq P\left(t\right)$ to organize the relative positions of the boxes in adjacent diagonals:

• if ${\epsilon }_j-{\epsilon }_i\in J$ and if $c\left({\text{box}}_j\right)\ne \ell -1$ or $c\left({\text{box}}_i\right)\ne 0,$ place ${\text{box}}_j$ northwest of ${\text{box}}_i;$
• if ${\epsilon }_j-{\epsilon }_i\notin J$ and if $c\left({\text{box}}_j\right)\ne \ell -1$ or $c\left({\text{box}}_i\right)\ne 0,$ place ${\text{box}}_j$ southeast of ${\text{box}}_i;$
• if ${\epsilon }_j-{\epsilon }_i\in J$ and $c\left({\text{box}}_j\right)=\ell -1$ or $c\left({\text{box}}_i\right)=0,$ place ${\text{box}}_j$ southeast of ${\text{box}}_i;$
• if ${\epsilon }_j-{\epsilon }_i\notin J$ and $c\left({\text{box}}_j\right)=\ell -1$ or $c\left({\text{box}}_i\right)=0,$ place ${\text{box}}_j$ northwest of ${\text{box}}_i\text{.}$

Thus, $t$ determines the number of boxes in each diagonal and $K$ determines the relative positions of the boxes in adjacent diagonals. This information completely determines the $\ell \text{-periodic}$ configuration of boxes associated to the pair $(t,J)\text{.}$

A $\ell \text{-periodic}$ standard tableau is an $\ell \text{-periodic}$ filling $p$ of the boxes with $1,2,\dots ,n$ such that

1. if $i<j$ and ${\text{box}}_i$ and ${\text{box}}_j$ are in the same diagonal then $p\left(i\right)<p\left(j\right),$
2. if $i<j$ and ${\text{box}}_i$ and ${\text{box}}_j$ are in adjacent diagonals with ${\text{box}}_j$ southwest of ${\text{box}}_i$ then $p\left(i\right)<p\left(j\right),$
3. if $i<j$ and ${\text{box}}_i$ and ${\text{box}}_j$ are in adjacent diagonals with ${\text{box}}_j$ northeast of ${\text{box}}_i$ then $p\left(i\right)>p\left(j\right),$

where $p\left(i\right)$ denotes the entry in ${\text{box}}_i\text{.}$ An $\ell \text{-periodic}$ standard tableau $p$ corresponds to a permutation in $S_n$ via the correspondence

$$\begin{array}{ccc}\left\{\text{standard tableaux}\right\}& \leftrightarrow & {ℱ}^{(t,J)},\\ p& ⟼& \left(\begin{array}{cccc}1& 2& \dots & n\\ p\left(1\right)& p\left(2\right)& \dots & p\left(n\right)\end{array}\right)\text{.}\end{array}$$

Example. Suppose $q^2=e^{2\pi i/4}$ and

$$t=(q^0,q^0,q^0,q^0,q^2,q^2,q^2,q^4,q^4,q^6,q^6,q^6,q^6,q^6)\text{.}$$

Then

$$\begin{array}{ccc}Z\left(t\right)& =& \{{\epsilon }_2-{\epsilon }_1,{\epsilon }_3-{\epsilon }_1,{\epsilon }_4-{\epsilon }_1,{\epsilon }_3-{\epsilon }_2,{\epsilon }_4-{\epsilon }_2,{\epsilon }_4-{\epsilon }_3,{\epsilon }_6-{\epsilon }_5,{\epsilon }_7-{\epsilon }_5,\dots \}\text{and}\\ P\left(t\right)& =& \{{\epsilon }_5-{\epsilon }_1,{\epsilon }_5-{\epsilon }_2,{\epsilon }_5-{\epsilon }_3,{\epsilon }_5-{\epsilon }_4,{\epsilon }_6-{\epsilon }_1,\dots ,{\epsilon }_{14}-{\epsilon }_9,{\epsilon }_{10}-{\epsilon }_1,\\ & & {\epsilon }_{10}-{\epsilon }_2,\dots ,{\epsilon }_{14}-{\epsilon }_4\}\text{.}\end{array}$$

If

$$\begin{array}{ccc}J& =& \{{\epsilon }_5-{\epsilon }_2,{\epsilon }_5-{\epsilon }_3,{\epsilon }_5-{\epsilon }_4,{\epsilon }_6-{\epsilon }_3,{\epsilon }_6-{\epsilon }_4,{\epsilon }_8-{\epsilon }_5,{\epsilon }_8-{\epsilon }_6,{\epsilon }_8-{\epsilon }_7,{\epsilon }_9-{\epsilon }_7,\\ & & {\epsilon }_{10}-{\epsilon }_9,{\epsilon }_{11}-{\epsilon }_9,{\epsilon }_{12}-{\epsilon }_9,{\epsilon }_{12}-{\epsilon }_2,{\epsilon }_{12}-{\epsilon }_3,{\epsilon }_{12}-{\epsilon }_4,{\epsilon }_{13}-{\epsilon }_2,{\epsilon }_{13}-{\epsilon }_3,\\ & & {\epsilon }_{13}-{\epsilon }_4,{\epsilon }_{14}-{\epsilon }_3,{\epsilon }_{14}-{\epsilon }_4\}\end{array}$$

then the corresponding $\ell \text{-periodic}$ configuration of boxes and a sample $\ell \text{-periodic}$ standard tableau are

$$\begin{array}{cc} 10 11 8 10 1 5 11 8 … … 12 1 5 9 13 2 6 12 14 3 9 13 2 6 4 7 14 3 4 7 & 3 3 2 3 0 1 3 2 … … 3 0 1 2 3 0 1 3 3 0 2 3 0 1 0 1 3 0 0 1 \\ \text{numbering of boxes}& \text{contents of boxes}\\ \end{array}$$ $$\begin{array}{c} 8 11 5 8 1 6 11 5 … … 2 1 6 3 7 9 12 2 10 13 3 7 9 12 14 4 10 13 14 4 \\ \text{a standard tableau}p\end{array}$$

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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