The Weyl group of type $D_n$

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: 28 May 2010

The Weyl group of type $D_n$

The Weyl group of type $D_n$ is the group of $n\times n$ matrices such that

1. there is exactly one nonzero entry in each row and in ech column,
2. the nonzero entries are $\pm 1,$
3. the product of the nonzero entries is 1.

Condition c) says that there are an even number of -1's in the matrix. The Weyl group $W{D}_n$ is a normal subgroup of index 2 in $W{B}_n$ and has order $2^{n-1}n!.$ The reflections in $W{D}_n$ are $$s_{{\epsilon }_i-{\epsilon }_j}=\left(i,j\right)\left(-i,-j\right),\text{ and }{s}_{{\epsilon }_i+{\epsilon }_j}=\left(i,-j\right)\left(-i,j\right),1\le i<j\le n.$$ The simple reflections in $W{D}_n$ are

$2\le i\le n.$

The Weyl group of type $D_n$ has a presentation given by generators $s_1,\dots ,s_n$ and relations corresponding to the Dynkin diagram $D_n$ and $s_i^2=1,$ for $1\le i\le n.$

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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