Indexings of canonical bases: Lyndon words, MV polytopes and the path model
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: 8 April
Affine Weyl group
Let Q=iIi
and*=iIi
with a symmetric bilinear form ,:**
given by values i,
so that A=iji=2iii is the Cartan matrix of a symmetrisable Kac-Moody Lie algebra .
Let +=positive roots of.
The Weyl group is
W0=si|iIGL*
with si:**-ii.
The affine Weyl group is
W=W0*Q=wX|wW0,Q
with X:**+.
The alcoves
are the connected components of *\+,j+jd+jd=*|+j=0.
Example:
*=+-,=++-
Each alcove has two types of address, wX
and si1,sil.
+=+-++-
since GL=+-+-
if +<-.
W
is generated by s0
and si,iI
where s0=Xs
and Walcoves
W0alcoves in the 0-hexagon
Qhexagons.
Chevalley groups G
G is generated by "elementary matrices" XfandX-f,f,+
with relations (see Steinberg of Parkinson-Schwer-Ram) where XfX--f-1Xf=hfn
and hfhf=h+f,for,Q.
The loop group
is Gt
where t=a-lt-l+a-l+1t-l+1+|ai,l.
Define X+jdc=X ctjt=ht-1,forQ,
n+jd=X+jd1X--jd-1X+jd1.
Let X0c=X-+dc,Xic=Xic,
n0=n-+dc,ni=ni.
Let [[t]]=a0+a1t+a2t2+|ai.
Example
I
=+-,A=2-1-12.
G=SL3
is generated by X+f=1f0010001X-f=10001f001X+-f=10f010001
X-+f=100f10001X--f=100010f01X-+--f=100010f01
Q=m++n-|m,n and t=hm++n-t-1=t-m000tm-n000tn,=m++n-.
X0c=X-+dc=X-ct=1000 0ct01,
n0=00-t-10 0t01n+=010-1 0001n-=1000 10-10
MV intersections and MV cycles
G=Gt|K=G[[t]]t=0G||I=B=Xchc|+,c,Q
G/K
is the loop Grassmannian.
G/I
is the affine flag variety.
G=wWIwIG=+KtK
G=vWU-vIG=U-tK
where U-=X-