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 A2, with rows indexed by the partitions 3,21,13 and columns indexed by the subsets ∅,1,2,12, D=100001100001 and DtD=100010010001100001100001=1000011001100001 For B2, with rows indexed by pairs of partitions, 2∅,12∅,11,∅2,∅12 and columns indexed by the subsets ∅,1,2,12, D==10000100011000100001 picture of bruhat graph

and DtD=1000001100001100000110000100011000100001=1000021001200001 For I25, with rows and columns indexed by χ1+,χ21,χ22,χ1-, and columns indexed by the subsets ∅,1,2,12, D=1000011001100001 picture of bruhat graph
and DtD=10000110011000011000011001100001=1000022002200001 For G2, with rows indexed by pairs of partitions, χ1++,χ1+-,χ21,χ22,χ1-+,χ1--, and columns indexed by the subsets ∅,1,2,12, D=100001000110011000100001 picture of bruhat graph
and DtD=100000011100001110000001100001000110011000100001=1000032002300001 So, in general, C=10000⌊m2⌋⌊m-12⌋00⌊m-12⌋⌊m2⌋00001 for I2m. If m is odd then C=10000m2m-1200m-12m-1200001 and D=10000110⋮⋮⋮⋮01100001 with rows indexed by χ1+,χ21,χ22,…,χ2m-12,χ1-, and columns indexed by the subsets ∅,1,2,12. If m is even then C=10000m2m-1200m-12m200001 and D=1000010001100110⋮⋮⋮⋮011000100001 with rows indexed by χ1++,χ1+-,χ21,χ22,…,χ2m-32,χ1-+,χ1--, and columns indexed by the subsets ∅,1,2,12.

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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