Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last updates: 16 February 2012
The corresponding dual canonical basis elements are
General formulas
If
for the Baumann-Gaussent formula
Then
If with
then
,
for ,
with
,
and
.
If with
then
,
for ,
with
,
and
.
Example: If with
and
,
then
,
the corresponding column strict tableau is
and the bead diagram is
and
,
and
.
Note that
so that the extremal terms correspond to paths near the boundary of the MV-polytope.
Example: If and , with
and
,
then
,
the corresponding column strict tableau is
and the bead diagram is
PICTURE OF BEAD DIAGRAM HERE
and
,
and
.
Notes and References
This analysis of MV-cycles and irreducible characters of quiver Hecke algebras for type
is part of joint work with A. Ghitza and S. Kannan on the relationship between MV-cycles and the Borel-Weil-Bott theorem. Of crucial importance here is the
result of Berenstein-Zelevinsky [BZ] REFERENCE HERE!!! characterizing the dual canonical basis for the
Type case.
References
[An]
J. Anderson,
A polytope calculus for semisimple groups, Duke Math. J.
116 (2003), 567-588.
MR1958098 (2004a:20047)
[MG]
S. Morier-Genoud,
Relèvement Géométrique de l'involution Schützenberger et applications,
Thèse de Doctorat, l'Université Claude Bernard - Lyon 1, June 2006.
[BZ]
A. Berenstein and A. Zelevinsky,
Canonical bases for the quantum group of type ,
and piecewise linear combinatorics,
Duke Math J. 143 (1996), 473-502.
[Ka]
M. Kashiwara,
On Crystal Bases, in Representations of groups (Banff 1994), pp. 155-197,
Canadian Math. Soc. Conf. Proc. 16 American Math. Soc. 1995.
MR1357199
[Km1]
J. Kamnitzer,
Mirković-Vilonen cycles and polytopes, Ann. Math. ???
MR??????
[Km2]
J. Kamnitzer,
The crystal structure on the set of Mirković-Vilonen polytopes, Adv. Math.
215 (2007), 66-93.
MR2354986