Two boundary Hecke Algebras and the combinatorics of type $C$

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

Last updated: 27 January 2015

Calibrated representations of $H_k^{\text{ext}}$

A calibrated $H_k^{\text{ext}}\text{-module}$ is an $H_k^{\text{ext}}\text{-module}$ $M$ such that $W_0,W_1,\dots ,W_k$ are simultaneously diagonalizable as operators on $M\text{.}$ In the context of (2.37), $M$ is calibrated if $$\begin{array}{cc}{\displaystyle M=\underset{\gamma \in 𝒞}{⨁}{M}_{\gamma },\text{where}{M}_{\gamma }=\{m\in M\hspace{0.17em}|\hspace{0.17em}{W}_{0}m=zm\hspace{0.17em}\text{and}\hspace{0.17em}{W}_{i}m={\gamma }_{i}m\hspace{0.17em}\text{for}\hspace{0.17em}i=1,\dots ,k\}}& \text{(3.1)}\end{array}$$ for $\gamma =(z,{\gamma }_1,\dots ,{\gamma }_k)\in 𝒞\text{.}$ Another formulation is that $M$ is calibrated if $M$ has a basis of simultaneous eigenvectors for $W_0,\dots ,W_k\text{.}$ This section follows the framework of [Ram2003] in developing combinatorial tools for describing the structure and the classification of irreducible calibrated $H_k^{\text{ext}}\text{-modules.}$ In Section 5 we will use this combinatorics to analyze and classify the $H_k^{\text{ext}}\text{-modules}$ arise in the basic Schur-Weyl duality settings.

With notations as in the definition of ${𝒲}_0$ in (2.2), the reflection representation of ${𝒲}_0$ is the action of ${𝒲}_0$ on ${𝔥}_{ℝ}={ℝ}^k$ given by $$w(c_1,\dots ,c_k)=(c_{w^{-1}\left(1\right)},\dots ,c_{w^{-1}\left(k\right)}),\text{where}\hspace{0.17em}{c}_{-i}=-c_i\hspace{0.17em}\text{for}\hspace{0.17em}i=1,2,\dots ,k\text{.}$$ The dual space ${𝔥}_{ℝ}^{*}$ has basis ${\epsilon }_1,\dots ,{\epsilon }_k,$ where ${\epsilon }_i:{𝔥}_{ℝ}\to ℝ$ is the $ℝ\text{-linear}$ map given by ${\epsilon }_i({\gamma }_1,\dots ,{\gamma }_k)={\gamma }_i\text{.}$ With ${\epsilon }_{-i}=-{\epsilon }_i,$ the action of ${𝒲}_0$ on ${ℝ}^k$ produces an action on ${𝔥}_{ℝ}^{*}$ given by $w{\epsilon }_i={\epsilon }_{w^{-1}\left(i\right)}\text{.}$

Let $$\begin{array}{rcl}{\displaystyle R^{+}}& =& {\displaystyle \{{\epsilon }_1,\dots ,{\epsilon }_k\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i,{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}1\le i<j\le k\}}\\ & =& {\displaystyle \{{\epsilon }_1,\dots ,{\epsilon }_k\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}1\le i<j\le k\}\bigsqcup \{{\epsilon }_j-{\epsilon }_{-i}\hspace{0.17em}|\hspace{0.17em}1\le i<j\le k\}}\\ & =& {\displaystyle \{{\epsilon }_1,\dots ,{\epsilon }_k\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}i,j\in \{-k,\dots ,-1,1,\dots ,k\},i<j,i\ne -j\}\text{.}}\end{array}$$ If $w\in {𝒲}_0,$ the inversion set of $w$ is $$\begin{array}{rcll}{\displaystyle R\left(w\right)}& =& {\displaystyle \{\alpha \in R^{+}\hspace{0.17em}|\hspace{0.17em}w\alpha \notin R^{+}\}}& \text{(3.2)}\\ & =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}\text{if}\hspace{0.17em}i>0\hspace{0.17em}\text{and}\hspace{0.17em}w\left(i\right)<0\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}\text{if}\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}w\left(i\right)>w\left(j\right)\}}& (3.3)\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}\text{if}\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}-w\left(i\right)>w\left(j\right)\}\text{.}}\end{array}$$ The chambers are the connected components of ${𝔥}_{ℝ}\backslash \bigcup _{\alpha \in R^{+}}{𝔥}^{\alpha },$ where ${𝔥}^{\alpha }=\{\gamma \in {𝔥}_{ℝ}\hspace{0.17em}|\hspace{0.17em}\alpha \left(\gamma \right)=0\}\text{.}$ The fundamental chamber in ${𝔥}_{ℝ}$ is $$C=\{\text{c}\in {𝔥}_{ℝ}\hspace{0.17em}|\hspace{0.17em}\alpha \left(\gamma \right)\in {ℝ}_{>0}\hspace{0.17em}\text{for}\hspace{0.17em}\alpha \in R^{+}\}=\{(c_1,\dots ,c_k)\in {ℝ}^k\hspace{0.17em}|\hspace{0.17em}0<c_1<c_2<\cdots <c_k\},$$ and the group ${𝒲}_0$ can be identified with the set of chambers via the bijection $$\begin{array}{ccc}{𝒲}_0& ⟷& \left\{\text{chambers}\right\}\\ w& ⟼& w^{-1}C\text{.}\end{array}\text{Since}{w}^{-1}C=\{\text{c}\in {𝔥}_{ℝ}\hspace{0.17em}|\hspace{0.17em}\genfrac{}{}{0ex}{}{\alpha \left(\text{c}\right)\in {ℝ}_{<0}\hspace{0.17em}\text{if}\hspace{0.17em}\alpha \in R\left(w\right)\hspace{0.17em}\text{and}}{\alpha \left(\text{c}\right)\in {ℝ}_{>0}\hspace{0.17em}\text{if}\hspace{0.17em}\alpha \in R^{+}\backslash R\left(w\right)}\},$$ the set $R\left(w\right)$ determines $w\text{.}$

Local regions

For $\gamma =({\gamma }_1,\dots ,{\gamma }_k)\in {\left({ℂ}^{\times }\right)}^k$ define $$\begin{array}{cc}{\displaystyle \begin{array}{rcl}{\displaystyle Z\left(\gamma \right)}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}{\gamma }_i=\pm 1\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,{\gamma }_i{\gamma }_j^{-1}=1\}\bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,{\gamma }_i{\gamma }_j=1\},}\\ {\displaystyle P\left(\gamma \right)}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}{\gamma }_i\in \{{\left(t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}}\right)}^{\pm 1},{(-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}})}^{\pm 1}\}\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,{\gamma }_i{\gamma }_j^{-1}=t^{\pm 1}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j,{\gamma }_i{\gamma }_j=t^{\pm 1}\}\text{.}}\end{array}}& \text{(3.4)}\end{array}$$ Using the conversion from ${\gamma }_i$ to $c_i$ as in (2.32), let $$\begin{array}{cc}{\displaystyle {\gamma }_i=-t^{c_i},\text{and}-t^{r_1}=-t_k^{\frac{1}{2}}{t}_0^{-\frac{1}{2}}\text{and}-t^{r_2}=t_k^{\frac{1}{2}}{t}_0^{\frac{1}{2}},}& \text{(3.5)}\end{array}$$ so that $-t^{\pm r_1}$ and $-t^{\pm r_2}$ are the eigenvalues of $W_1$ that cause ${\tau }_0^2$ to have a nonzero kernel (see (2.42)). Then, for $\text{c}=(c_1,\dots ,c_k)\in {ℂ}^k$ let $c_{-i}=-c_i$ and define $$\begin{array}{rcll}{\displaystyle Z\left(\text{c}\right)}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}{c}_i=0\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{c}_j-c_i=0\}\bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{c}_j+c_i=0\}}& \text{(3.6)}\\ {\displaystyle P\left(\text{c}\right)}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}{c}_i\in \{\pm r_1,\pm r_2\}\}\bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{c}_j-c_i=\pm 1\}}& \text{(3.7)}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{c}_j+c_i=\pm 1\}\text{.}}\end{array}$$ A local region is a pair $(\text{c},J)$ with $\text{c}\in {ℂ}^k$ and $J\subseteq P\left(\text{c}\right)\text{.}$ The set of standard tableaux of shape $(\text{c},J)$ is $$\begin{array}{cc}{\displaystyle {ℱ}^{(\text{c},J)}=\{w\in {𝒲}_0\hspace{0.17em}|\hspace{0.17em}R\left(w\right)\cap Z\left(\text{c}\right)=\varnothing ,R\left(w\right)\cap P\left(\text{c}\right)=J\}\text{.}}& \text{(3.8)}\end{array}$$

As in [Ram2003, §5 and §8] the local regions $(\text{c},J)$ and standard tableaux $w\in {ℱ}^{(\text{c},J)}$ can be converted to configurations of boxes $\kappa$ and standard tableaux $S$ of shape $\kappa$ similar to those that are familiar in the literature on irreducible representations of Weyl groups of classical types. As explained in [Ram2003, §5.11], the definitions of $Z\left(\text{c}\right)$ and $P\left(\text{c}\right)$ make it possible to view the general case $\text{c}\in {ℂ}^k$ as pieced together from the cases $\text{c}\in {(\mathbb{Z}+\beta )}^k$ where $\beta$ runs over a set of representatives of the $\mathbb{Z}\text{-cosets}$ in $ℂ\text{.}$ Below we make the conversion between local regions and configurations of boxes explicit for the cases when $\text{c}\in {\mathbb{Z}}^k$ and $\text{c}\in {(\mathbb{Z}+\frac{1}{2})}^k\text{.}$ These are the cases that appear in the Schur-Weyl duality approach to the representations of $H_k^{\text{ext}}$ which we explore in Section 5. As in [Ram2003, §8], it is also true that these cases are sufficient to determine the general $\text{c}\in {(\mathbb{Z}+\beta )}^k$ setting but we shall not do this in detail here.

Let $(\text{c},J)$ be a local region with $\text{c}=(c_1,\dots ,c_k),$ $$\begin{array}{cc}{\displaystyle \text{c}\in {\mathbb{Z}}^k\text{or}\text{c}\in {(\mathbb{Z}+\frac{1}{2})}^k,\text{and}0\le c_1\le \cdots \le c_k\text{.}}& \text{(3.9)}\end{array}$$ Start with an infinite arrangement of NW to SE diagonals, numbered consecutively from $\mathbb{Z}$ or $\mathbb{Z}+\frac{1}{2},$ increasing southwest to northeast (see Example 1). The configuration $\kappa$ of boxes corresponding to the local region $(\text{c},J)$ has $2k$ boxes (labeled ${\text{box}}_{-k},\dots ,{\text{box}}_{-1},{\text{box}}_1,\dots ,{\text{box}}_k\text{)}$ with the following conditions.

$\left(\kappa 1\right)$	Location: ${\text{box}}_i$ is on diagonal $c_i\text{,}$ where $c_{-i}=-c_i$ for $i\in \{-k,\dots ,-1\}\text{.}$
$\left(\kappa 2\right)$	Same diagonals: ${\text{box}}_i$ is NW of ${\text{box}}_j$ if $i<j$ and ${\text{box}}_i$ and ${\text{box}}_j$ are on the same diagonal.
$\left(\kappa 3\right)$	Adjacent diagonals: If ${\epsilon }_j-{\epsilon }_i\in J,$ then ${\text{box}}_j$ is NW (strictly north and weakly west) of ${\text{box}}_i\text{:}$ $\begin{array}{c} j i \end{array}$ If ${\epsilon }_j-{\epsilon }_i\in P\left(\text{c}\right)-J,$ then ${\text{box}}_j$ is SE (weakly south and strictly east) of ${\text{box}}_i\text{:}$ $\begin{array}{c} i j \end{array}$
$\left(\kappa 4\right)$	Markings: There is a marking on each of the diagonals $r_1\text{,}$ $-r_1,$ $r_2$ and $-r_2\text{.}$ If ${\epsilon }_i\in J,$ ${\text{box}}_i$ is NW of the marking in diagonal $c_i\text{:}$ $\begin{array}{c} i \end{array}$ If ${\epsilon }_i\in P\left(\text{c}\right)-J,$ then ${\text{box}}_i$ is SE of the marking in diagonal $c_i\text{:}$ $\begin{array}{c} i \end{array}$

Condition $\left(\kappa 1\right)$ enables the values $(c_{-k},\dots ,c_{-1},c_1,\dots ,c_k)$ to be read off of configuration $\kappa \text{.}$ The sets $Z\left(\text{c}\right)\text{,}$ $P\left(\text{c}\right)\text{,}$ and $J$ can also be determined from the configuration $\kappa$ since $$\begin{array}{rcl}{\displaystyle Z\left(\text{c}\right)}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}0<i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{is in diagonal 0}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{are in the same diagonal}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{are both in diagonal 0}\},}\\ {\displaystyle P\left(\text{c}\right)}& =& {\displaystyle \left\{{\epsilon }_i\hspace{0.17em}\right|\hspace{0.17em}0<i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{is in diagonal}\hspace{0.17em}{r}_1\hspace{0.17em}\text{or}\hspace{0.17em}{r}_2\},}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j-{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{are in adjacent diagonals}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\hspace{0.17em}|\hspace{0.17em}0<i<j\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_{-i}\hspace{0.17em}\text{and}\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{are in adjacent diagonals}\},\text{and}}\\ {\displaystyle J}& =& {\displaystyle \{{\epsilon }_i\in P\left(\text{c}\right)\hspace{0.17em}|\hspace{0.17em}{\text{box}}_i\hspace{0.17em}\text{is NW of the marking}\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j-{\epsilon }_i\in P\left(\text{c}\right)\hspace{0.17em}|\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is northwest of}\hspace{0.17em}{\text{box}}_i\}}\\ & & {\displaystyle \bigsqcup \{{\epsilon }_j+{\epsilon }_i\in P\left(\text{c}\right)\hspace{0.17em}|\hspace{0.17em}{\text{box}}_j\hspace{0.17em}\text{is northwest of}\hspace{0.17em}{\text{box}}_{-i}\}\text{.}}\end{array}$$ A standard filling of the boxes of $\kappa$ is a bijective function $S:\kappa \to \{-k,\dots ,-1,1,\dots ,k\}$ such that

(S1)	Symmetry: $S\left({\text{box}}_{-i}\right)=-S\left({\text{box}}_i\right)\text{.}$
(S2)	Same diagonals: If $0<i<j$ and ${\text{box}}_i$ and ${\text{box}}_j$ are on the same diagonal then $S\left({\text{box}}_i\right)<S\left({\text{box}}_j\right)\text{.}$
(S3)	Adjacent diagonals: If $0<i<j,$ ${\text{box}}_i$ and ${\text{box}}_j$ are on adjacent diagonals, and ${\text{box}}_j$ is NW of ${\text{box}}_i\text{,}$ then $S\left({\text{box}}_j\right)<S\left({\text{box}}_i\right)\text{.}$ If $0<i<j,$ ${\text{box}}_i$ and ${\text{box}}_j$ are on adjacent diagonals, and ${\text{box}}_j$ is SE of ${\text{box}}_i\text{,}$ then $S\left({\text{box}}_j\right)>S\left({\text{box}}_i\right)\text{.}$
(S4)	Markings: If ${\text{box}}_i$ is on a marked diagonal and is SE of the marking, then $S\left({\text{box}}_i\right)>0\text{.}$ If ${\text{box}}_i$ is on a marked diagonal and is NW of the marking, then $S\left({\text{box}}_i\right)<0\text{.}$

The identity filling of a configuration $\kappa$ is the filling $F$ of the boxes of $\kappa$ given by $F\left({\text{box}}_i\right)=i,$ for $i=-k,\dots ,-1,1,\dots ,k\text{.}$ The identity filling of $\kappa$ is usually not a standard filling of $\kappa$ (see Example 1).

Let $k=4,$ $r_1=1,$ and $r_2=3\text{.}$ Consider $\text{c}=(-3,-2,-2,2,2,3)\text{.}$ Then $$Z\left(\text{c}\right)=\{{\epsilon }_1-{\epsilon }_2\}\text{and}P\left(\text{c}\right)=\{{\epsilon }_3,{\epsilon }_1-{\epsilon }_3,{\epsilon }_2-{\epsilon }_3\}\text{.}$$ The box configurations corresponding to $J=\{{\epsilon }_2-{\epsilon }_3\}$ and $J=\{{\epsilon }_3,{\epsilon }_1-{\epsilon }_3,{\epsilon }_2-{\epsilon }_3\}$ (filled with their identity fillings) are $$\begin{array}{ccc}\begin{array}{c} 1 3 2 -2 -3 -1 0 1 2 3 4 5 -1 -2 -3 -4 -5 \end{array}& & \begin{array}{c} 3 1 2 -2 -1 -3 0 1 2 3 4 5 -1 -2 -3 -4 -5 \end{array}\\ J=\{{\epsilon }_2-{\epsilon }_3\}& & J=\{{\epsilon }_3,{\epsilon }_1-{\epsilon }_3,{\epsilon }_2-{\epsilon }_3\}\end{array}$$ For both configurations, the identity filling is not a standard filling. Examples of standard fillings of the configuration corresponding to $J=\{{\epsilon }_2-{\epsilon }_3\}$ include $$\begin{array}{c} 1 2 3 -3 -2 -1 \end{array},,\begin{array}{c} -1 2 3 -3 -2 -1 \end{array},\text{and}\begin{array}{c} -2 1 3 -3 -1 -2 \end{array},\text{but not}\begin{array}{c} -3 -2 1 -1 2 3 \end{array}\text{.}$$

The proof of the following proposition is a straightforward, though slightly tedious, check that the conditions $R\left(w\right)\cap Z\left(\text{c}\right)=\varnothing$ and $R\left(w\right)\cap P\left(\text{c}\right)=J$ from (3.8) convert to the conditions (S2), (S3), (S4) on standard fillings of shape $\kappa \text{.}$ The proof is similar to the proof of [Ram2003, Thm. 5.9].

Let $\kappa$ be a configuration of boxes corresponding to a local region $(\text{c},J)$ with $\text{c}\in {\mathbb{Z}}^k$ or $\text{c}\in {(\mathbb{Z}+\frac{1}{2})}^k\text{.}$ For $w\in {𝒲}_0$ let $S_w$ be the filling of the boxes of $\kappa$ given by $$S_w\left({\text{box}}_i\right)=w\left(i\right),\text{for}\hspace{0.17em}i=-k,\dots ,-1,1,\dots k\text{.}$$ The map $$\begin{array}{ccc}{ℱ}^{(\text{c},J)}& ⟶& \left\{\text{standard fillings}\hspace{0.17em}S\hspace{0.17em}\text{of the boxes of}\hspace{0.17em}\kappa \right\}\\ w& ⟼& S_w\end{array}\text{is a bijection.}$$

Let $k=12,$ $r_1=\frac{3}{2},$ $r_2=\frac{15}{2},$ $\text{c}=(\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{3}{2},\frac{5}{2},\frac{9}{2},\frac{11}{2},\frac{13}{2},\frac{13}{2},\frac{15}{2},\frac{15}{2},\frac{17}{2})$ and $$J=\left\{\genfrac{}{}{0ex}{}{{\epsilon }_3,{\epsilon }_{10},{\epsilon }_3-{\epsilon }_2,{\epsilon }_4-{\epsilon }_2,{\epsilon }_5-{\epsilon }_4,{\epsilon }_8-{\epsilon }_7,\hfill }{{\epsilon }_{10}-{\epsilon }_8,{\epsilon }_{10}-{\epsilon }_9,{\epsilon }_{11}-{\epsilon }_9,{\epsilon }_{12}-{\epsilon }_{10},{\epsilon }_{12}-{\epsilon }_{11}\hfill }\right\}$$ Let $$w=\left(\begin{array}{cccccccccccc}1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12\\ -9& 10& -8& 7& 6& 3& 4& 1& 5& -11& 2& -12\end{array}\right)\in {ℱ}^{(\text{c},J)}\text{.}$$ Then, for the corresponding configuration of boxes $\kappa \text{,}$ the identity filling $F\text{,}$ and the standard filling $S_w$ corresponding to $w$ are $$F=\begin{array}{c} 12 10 8 11 6 7 9 -2 1 3 5 -4 4 -5 -3 -1 2 -9 -7 -6 -11 -8 -10 -12 172 152 12 -12 -152 -172 \end{array}\text{and}{S}_w=\begin{array}{c} -12 -11 1 2 3 4 5 -10 -9 -8 6 -7 7 -6 8 9 10 -5 -4 -3 -2 -1 11 12 172 152 12 -12 -152 -172 \end{array}$$

Borrowing a physical intuition, configurations are invariant under sliding boxes along diagonals like beads on an abacus, so long as boxes that run into each other are not allowed to exchange places, i.e. for most $c\in \mathbb{Z},$ $$\begin{array}{c} c+1 c \end{array}=\begin{array}{c} c+1 c \end{array}\ne \begin{array}{c} c+1 c \end{array}\text{.}$$ Then by arranging configurations so that the boxes are packed together, standard fillings of configurations are exactly analogous to standard tableaux for partitions.

The only exception to this physical intuition is for boxes on the diagonals $\pm \frac{1}{2}\text{.}$ Note that if $c_i=\frac{1}{2},$ then ${\text{box}}_i$ and ${\text{box}}_{-i}$ are on adjacent diagonals. However, since $2{\epsilon }_i={\epsilon }_i-{\epsilon }_{-i}\notin R^{+}$ and therefore never in $P\left(\text{c}\right)\text{,}$ the relative positions of ${\text{box}}_i$ and ${\text{box}}_{-i}$ will never be recorded in the set $J\text{.}$ For example, in Figure 2, the point where $(c_1,c_2)=(\frac{1}{2},\frac{1}{2})$ has two configurations, each with two boxes overlapping in indication that ${\text{box}}_i$ and ${\text{box}}_{-i}$ may "slide past each other". The drawing (with boxes filled in the identity filling) $$\begin{array}{c} \end{array}\text{represents the equivalence of}\begin{array}{c} -12 12 -2 -1 1 2 \end{array}\text{and}\begin{array}{c} -12 12 -2 1 -1 2 \end{array},$$ where ${\text{box}}_1$ and ${\text{box}}_{-1}$ can move freely past each other, and $$\begin{array}{c} \end{array}\text{represents the equivalence of}\begin{array}{c} 12 -12 2 1 -1 -2 \end{array}\text{and}\begin{array}{c} 12 -12 2 -1 1 -2 \end{array},$$ where ${\text{box}}_2$ and ${\text{box}}_{-2}$ can move freely past each other. In these two examples ${\epsilon }_1-{\epsilon }_{-2}\in P\left(\text{c}\right)$ and ${\epsilon }_2-{\epsilon }_{-1}\in P\left(\text{c}\right)$ and so the relative orientation of ${\text{box}}_2$ and ${\text{box}}_{-1}$ and the relative orientation of ${\text{box}}_1$ and ${\text{box}}_{-2}$ are recorded in $J\text{.}$ Each configuration has exactly two standard fillings.

Classifying and constructing calibrated representations

Theorem 3.3 below provides an indexing of the calibrated irreducible $H_k^{\text{ext}}\text{-modules}$ by skew local regions. A skew local region is a local region $(\text{c},J)\text{,}$ $\text{c}=(c_1,\dots ,c_k),$ such that if $w\in {ℱ}^{(\text{c},J)}$ then $w\text{c}=({\left(w\text{c}\right)}_1,\dots ,{\left(w\text{c}\right)}_n)$ satisfies $$\begin{array}{cc}{\displaystyle \begin{array}{c}{\left(w\text{c}\right)}_1\ne 0,{\left(w\text{c}\right)}_2\ne 0,{\left(w\text{c}\right)}_1\ne -{\left(w\text{c}\right)}_2,\\ {\left(w\text{c}\right)}_i\ne {\left(w\text{c}\right)}_{i+1}\hspace{0.17em}\text{for}\hspace{0.17em}i=1,\dots ,k-1,\text{and}{\left(w\text{c}\right)}_i\ne {\left(w\text{c}\right)}_{i+2}\hspace{0.17em}\text{for}\hspace{0.17em}i=1,\dots ,k-2\text{.}\end{array}}& \text{(3.10)}\end{array}$$ Theorem 3.3 is completely analogous to the same theorem for the case $t^{\frac{1}{2}}=t_0^{\frac{1}{2}}=t_k^{\frac{1}{2}}$ in [Ram2003, Theorem 3.5]. As explained in the discussion and remarks before [Ram2003, Lemma 3.1] in [Ram2003, §3], this depends on getting exactly the right definition of skew local region, which is accomplished by a detailed computation of the irreducible representations in rank two cases. More specifically, for $I\subseteq \{0,\dots ,k\},$ let $H_I$ be the subalgebra of $H_k^{\text{ext}}$ generated by ${\left\{T_i\right\}}_{i\in I}$ and $ℂ[W_1^{\pm 1},\dots ,W_k^{\pm 1}]\text{.}$ Then the conditions in (3.10) guarantee that for $w\in {ℱ}^{(\text{c},J)}$ and $i,j\in \{0,1,\dots ,k-1\},$ $$\text{there exists a calibrated}\hspace{0.17em}{H}_{\{i,j\}}\text{-module}\hspace{0.17em}M\hspace{0.17em}\text{with}\hspace{0.17em}{M}_{w\text{c}}^{\text{gen}}\ne 0\text{.}$$ For the cases where $H_{\{i,j\}}$ is of type $A\times A_1$ or of type $A_2$ this is checked in [Ram2002]. However, when $H_{\{i,j\}}$ is of type $C_2$ and there are three distinct parameters we do not know a reference for this and so, in the effort to provide a more complete presentation, we have done the appropriate analysis in Section 4 for all generic choices of the three parameters $t^{\frac{1}{2}},$ $t_0^{\frac{1}{2}},$ and $t_k^{\frac{1}{2}},$ as defined in (??), which we rewrite here: $t^{\frac{1}{2}}$ is not a root of unity and $$t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}},-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}}\notin \{1,-1,t^{\pm \frac{1}{2}},-t^{\pm \frac{1}{2}},t^{\pm 1},-t^{\pm 1}\}\text{and}{t}_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}}\ne {(-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}})}^{\pm 1}\text{.}$$

Assume $t^{\frac{1}{2}},$ $t_0^{\frac{1}{2}},$ and $t_k^{\frac{1}{2}}$ are invertible, $t^{\frac{1}{2}}$ is not a root of unity, and $$t_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}},-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}}\notin \{1,-1,t^{\pm \frac{1}{2}},-t^{\pm \frac{1}{2}},t^{\pm 1},-t^{\pm 1}\}\text{and}{t}_0^{\frac{1}{2}}{t}_k^{\frac{1}{2}}\ne {(-t_0^{-\frac{1}{2}}{t}_k^{\frac{1}{2}})}^{\pm 1}\text{.}$$

(a)

Let $(\text{c},J),$ $\text{c}=(c_1,\dots ,c_k),$ be a skew local region and let $z\in {ℂ}^{\times }\text{.}$ Define $$\begin{array}{cc}{\displaystyle H_k^{(z,\text{c},J)}={\text{span}}_{ℂ}\left\{v_w\hspace{0.17em}\right|\hspace{0.17em}w\in {ℱ}^{(\text{c},J)}\}}& \text{(3.11)}\end{array}$$ so that the symbols $v_w$ are a labeled basis of the vector space $H_k^{(z,\text{c},J)}\text{.}$ Let $${\gamma }_i=-t^{c_i}\text{for}\hspace{0.17em}i=1,2,\dots ,k,\text{and}{\gamma }_0=z{\gamma }_{w^{-1}\left(1\right)}^{-1}\cdots {\gamma }_{w^{-1}\left(k\right)}^{-1}\text{.}$$ Then the following formulas make $H_k^{(z,\text{c},J)}$ into an irreducible $H_k^{\text{ext}}\text{-module:}$ $$\begin{array}{cc}{\displaystyle P{W}_1\cdots W_k{v}_w=z{v}_w,P{v}_w={\gamma }_0{v}_w,W_i{v}_w={\gamma }_{w^{-1}\left(i\right)}{v}_w,}& \text{(3.12)}\end{array}$$ $$\begin{array}{rcll}{\displaystyle T_i{v}_w}& =& {\displaystyle {\left[T_i\right]}_{ww}{v}_w+\sqrt{-({\left[T_i\right]}_{ww}-t^{\frac{1}{2}})({\left[T_i\right]}_{ww}+t^{-\frac{1}{2}})}{v}_{s_{i}w},\text{for}\hspace{0.17em}i=1,\dots ,k-1,}& \text{(3.13)}\\ {\displaystyle T_0{v}_w}& =& {\displaystyle {\left[T_0\right]}_{ww}{v}_w+\sqrt{-({\left[T_0\right]}_{ww}-t_0^{\frac{1}{2}})({\left[T_0\right]}_{ww}+t_0^{-\frac{1}{2}})}{v}_{s_{0}w},}& \text{(3.14)}\end{array}$$ where $v_{s_{i}w}=0$ if $s_{i}w\notin {ℱ}^{(\text{c},J)}$ and $$\begin{array}{cc}{\displaystyle {\left[T_i\right]}_{ww}=\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-{\gamma }_{w^{-1}\left(i\right)}{\gamma }_{w^{-1}(i+1)}^{-1}},\text{and}{\left[T_0\right]}_{ww}=\frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}}){\gamma }_{w^{-1}\left(1\right)}^{-1}}{1-{\gamma }_{w^{-1}\left(1\right)}^{-2}}\text{.}}& \text{(3.15)}\end{array}$$

(b)

The map $$\begin{array}{ccc}{ℂ}^{\times }\times \left\{\text{skew local regions}\hspace{0.17em}(\text{c},J)\right\}& ⟷& \left\{\text{irreducible calibrated}\hspace{0.17em}{H}_k^{\text{ext}}\text{-modules}\right\}\\ (z,\text{c},J)& ⟼& H_k^{(z,\text{c},J)}\end{array}$$ is a bijection.

		Proof.
	This result follows from [Ram2003, Theorems 3.2 and 3.5]. It is only necessary to establish that the formulas in (3.12), (3.13) and (3.14) are correct. These are derived in a similar manner to [Ram2003, Proposition 3.3] as follows. As in [Ram2003, Theorem 3.2], if $M$ is an irreducible calibrated $H_k^{\text{ext}}\text{-module}$ then $$M=\underset{w\in {𝒲}_0}{⨁}{M}_{w\gamma }^{\text{gen}},\text{with}\hspace{0.17em}\text{dim}\left(M_{w\gamma }^{\text{gen}}\right)=1\hspace{0.17em}\text{if}\hspace{0.17em}{M}_{w\gamma }^{\text{gen}}\ne 0\text{.}$$ For $w\in {𝒲}_0,$ if $M_{w\gamma }^{\text{gen}}\ne 0,$ let $v_w$ be a nonzero vector in $M_{w\gamma }^{\text{gen}}\text{;}$ otherwise if $M_{w\gamma }^{\text{gen}}=0,$ let $v_w=0\text{.}$ By (2.38), ${\tau }_i{v}_w={\left[T_i\right]}_{s_{i}w,w}{v}_{s_{i}w}$ for some constant ${\left[T_i\right]}_{s_{i}w,w}$ and the definition of ${\tau }_i$ in (2.35) gives that $$\begin{array}{cc}{\displaystyle T_i{v}_w=\frac{t^{\frac{1}{2}}-t^{-\frac{1}{2}}}{1-{\gamma }_{w^{-1}\left(i\right)}{\gamma }_{w^{-1}(i+1)}^{-1}}{v}_w+{\left[T_i\right]}_{s_{i}w,w}{v}_{s_{i}w},\text{for}\hspace{0.17em}i=1,\dots ,k,}& \text{(3.16)}\end{array}$$ and $$\begin{array}{cc}{\displaystyle T_0{v}_{\gamma }=\frac{(t_0^{\frac{1}{2}}-t_0^{-\frac{1}{2}})+(t_k^{\frac{1}{2}}-t_k^{-\frac{1}{2}}){\gamma }_{w^{-1}\left(1\right)}^{-1}}{1-{\gamma }_{w^{-1}\left(1\right)}^{-2}}{v}_w+{\left[T_0\right]}_{s_{0}w,w}{v}_{s_{0}w}\text{.}}& \text{(3.17)}\end{array}$$ Thus $T_0$ is an operator on the subspace ${\text{span}}_{ℂ}\{v_w,v_{s_{0}w}\}$ satisfying $(T_0-t_0^{\frac{1}{2}})(T_0+t^{-\frac{1}{2}})=0$ by (H). Restricting to the action on ${\text{span}}_{ℂ}\{v_w,v_{s_{0}w}\},$ the formulas in (3.14) now follow from the following argument about general $2\times 2$ matrices. If a $2\times 2$ matrix $\left[T_0\right]$ has eigenvalues ${\alpha }_1$ and ${\alpha }_2,$ $$\left[T_0\right]=\left(\begin{array}{cc}{\left[T_0\right]}_{ww}& {\left[T_0\right]}_{w,s_{0}w}\\ {\left[T_0\right]}_{s_{0}w,w}& {\left[T_0\right]}_{s_{0}w,s_{0}w}\end{array}\right),\text{then}(\left[T_0\right]-{\alpha }_1)(\left[T_0\right]-{\alpha }_2)=0$$ is the characteristic polynomial for $\left[T_0\right],$ and it follows that $$\begin{array}{rcl}{\displaystyle \text{Tr}\left(\left[T_0\right]\right)}& =& {\displaystyle {\left[T_0\right]}_{ww}+{\left[T_0\right]}_{s_{0}w,s_{0}w}={\alpha }_1+{\alpha }_2,\text{and}}\\ {\displaystyle \text{det}\left(\left[T_0\right]\right)}& =& {\displaystyle {\left[T_0\right]}_{ww}{\left[T_0\right]}_{s_{0}w,s_{0}w}-{\left[T_0\right]}_{w,s_{0}w}{\left[T_0\right]}_{s_{0}w,w}={\alpha }_1{\alpha }_2\text{.}}\end{array}$$ Thus $$\begin{array}{rcl}{\displaystyle -{\left[T_0\right]}_{w,s_{0}w}{\left[T_0\right]}_{s_{0}w,w}}& =& {\displaystyle {\alpha }_1{\alpha }_2-{\left[T_0\right]}_{ww}{\left[T_0\right]}_{s_{0}w,s_{0}2}={\alpha }_1{\alpha }_2-{\left[T_0\right]}_{ww}(({\alpha }_1+{\alpha }_2)-{\left[T_0\right]}_{ww})}\\ & =& {\displaystyle {\alpha }_1{\alpha }_2-({\alpha }_1+{\alpha }_2){\left[T_0\right]}_{ww}+{\left({\left[T_0\right]}_{ww}\right)}^2=({\left[T_0\right]}_{ww}-{\alpha }_1)({\left[T_0\right]}_{ww}-{\alpha }_2)\text{.}}\end{array}$$ Choosing a normalization of $v_{s_{0}w}$ so that the matrix of $\left[T_0\right]$ is symmetric, we have ${\left[T_0\right]}_{w,s_{0}w}={\left[T_0\right]}_{s_{0}w,w}$ and $${\left[T_0\right]}_{s_{0}w,w}=\sqrt{{\left({\left[T_0\right]}_{s_{0}w,w}\right)}^2}=\sqrt{{\left[T_0\right]}_{w,s_{0}w}{\left[T_0\right]}_{w,s_{0}w}}=\sqrt{-({\left[T_0\right]}_{ww}-{\alpha }_1)({\left[T_0\right]}_{ww}-{\alpha }_2)}\text{.}$$ $\square$

Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type $C$ Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).

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