The tower <math> <msub> <mover> <mi>A</mi><mo>^</mo> </mover><mi>k</mi> </msub><mfenced> <mi>r</mi><mi>p</mi><mi>n</mi> </mfenced> </math>

The tower A^ k rpn

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 20 June 2010

The tower A^ k rpn

Let 1n be the trivial representation of G= G r,1n and let V=-span v1 vn be the reflection representation. Let G= G r,p,n = G r,1,n G r,p,n   and   L n-1 = G r,1,n-1 × /r G r,p,n . The G r,1,n-1 × /r module χ= 1 n-1 χ1 ,where   χ1 : /r * ξ ξ is, by restriction, an L n-1 module and for any G -module M, Ind L n-1 G Res L n-1 G M χ M Ind L n-1 G χ MV, where the first isomorphism comes from the tensor identity,

Ind L n-1 G Res L n-1 G M N ~ M Ind L n-1 G N g mn gm gn ,
for gG,mM,nN, and the fact that Ind L n-1 G W =G L n-1 W. Iterating (????) it follows that
Ind L n-1 G Res L n-1 G M χ k 1 V k   and   Res L n-1 G Ind L n-1 G Res L n-1 G k 1 V k
as G modules and L n-1 -modules respectively.

This analysis allows us to build the Bratteli diagram of Ak r1n . This graph is constructed inductively as followsL A^ 0 r1n = n . If k >0 then there are edges A^ k r1n : λ A^ k+ 1 2 : μ i   if  μ  is obtained from  λ  by removing a box from   λ i , and A^ k r1n : ν i A^ k+ 1 2 : γ   if  γ  is obtained from  ν  by adding a box to   ν i+1 , where we make the convention that μ r = μ 0 .

The Bratteli diagram of the algebra A k,r,p n

Recall that the simple G r,1,n modules are given by G r,1,n λ =span vT | T  is a standard tableau of shape  λ with action ti vT =s T i vT   and   si vT = si TT vT + 1+ si TT v si T . Then /p acts on the G r,1,n modules by σ: G r,1,n λ G r,1,n λ vT v σT and this action lifts to an action of /p on G r,1,n by automorphisms σ: G r,1,n λ G r,1,n λ tλ w ζ λ tλ w where  ζ= e 2πi/p . Then V k is a G r,1,n module and we can twist the action by any automorphism. So ti vj = ξζ vi , if  j=i, vi , if  ji. Then σ: V k σ* V k as G r,1,n modules and this operation commutes with the G r,p,n action. So σ End G r,p,n V k .

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)

page history