The 0 Hecke algebra (examples)

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

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 10 June 2010

The 0 Hecke algebra (examples)

For $A_2,$ with rows indexed by the partitions $\left(3\right),\left(21\right),\left(1^3\right)$ and columns indexed by the subsets $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\},$ $$D=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ and $$D^{t}D=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ & 0& 1& 0\\ & 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ For $B_2,$ with rows indexed by pairs of partitions, $$\left(\left(2\right),\varnothing \right),\left(\left(1^2\right)\varnothing \right),\left(\left(1\right),\left(1\right)\right),\left(\varnothing ,\left(2\right)\right),\left(\varnothing ,\left(1^2\right)\right)$$ and columns indexed by the subsets $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\},$ $$D==\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ picture of bruhat graph

and $$D^{t}D=\left(\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 1& 0& 0\\ 0& 0& 1& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 2& 1& 0\\ 0& 1& 2& 0\\ 0& 0& 0& 1\end{array}\right)$$ For $I_2\left(5\right),$ with rows and columns indexed by $${\chi }_1^{+},{\chi }_2^1,{\chi }_2^2,{\chi }_1^{-},$$ and columns indexed by the subsets $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\},$ $$D=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ picture of bruhat graph
and $$D^{t}D=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 2& 2& 0\\ 0& 2& 2& 0\\ 0& 0& 0& 1\end{array}\right)$$ For $G_2,$ with rows indexed by pairs of partitions, $${\chi }_1^{++},{\chi }_1^{+-},{\chi }_2^1,{\chi }_2^2,{\chi }_1^{-+},{\chi }_1^{--},$$ and columns indexed by the subsets $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\},$ $$D=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ picture of bruhat graph
and $$D^{t}D=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 1& 1& 1& 0& 0\\ 0& 0& 1& 1& 1& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 3& 2& 0\\ 0& 2& 3& 0\\ 0& 0& 0& 1\end{array}\right)$$ So, in general, $$C=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \lfloor \frac{m}{2}\rfloor & \lfloor \frac{m-1}{2}\rfloor & 0\\ 0& \lfloor \frac{m-1}{2}\rfloor & \lfloor \frac{m}{2}\rfloor & 0\\ 0& 0& 0& 1\end{array}\right)$$ for $I_2\left(m\right).$ If $m$ is odd then $$C=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \frac{m}{2}& \frac{m-1}{2}& 0\\ 0& \frac{m-1}{2}& \frac{m-1}{2}& 0\\ 0& 0& 0& 1\end{array}\right)$$ and $$D=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 1& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 1& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ with rows indexed by $${\chi }_1^{+},{\chi }_2^1,{\chi }_2^2,\dots ,{\chi }_2^{\frac{m-1}{2}},{\chi }_1^{-},$$ and columns indexed by the subsets $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\}.$ If $m$ is even then $$C=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \frac{m}{2}& \frac{m-1}{2}& 0\\ 0& \frac{m-1}{2}& \frac{m}{2}& 0\\ 0& 0& 0& 1\end{array}\right)$$ and $$D=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 1& 1& 0\\ 0& 1& 1& 0\\ ⋮& ⋮& ⋮& ⋮\\ 0& 1& 1& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)$$ with rows indexed by $${\chi }_1^{++},{\chi }_1^{+-},{\chi }_2^1,{\chi }_2^2,\dots ,{\chi }_2^{\frac{m-3}{2}},{\chi }_1^{-+},{\chi }_1^{--},$$ and columns indexed by the subsets $\varnothing ,\left\{1\right\},\left\{2\right\},\left\{1,2\right\}.$

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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