MV polytopes

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

Last updates: 10 February 2011

MV polytopes

A MV-polytope is b=Conv{ μw | wW } the convex hull of its vertices.

μ1 is the type of b,     and     μw0 is the weight of b.
A reduced word w0 = si1 siN for w0 induces an ordering on the positive coroots
β1 = αi1 , β2 = si1 αi2 , βN = si1 siN-1 αiN . (rootorder)
A multisegment is a sequence (1 ,, N) (0) N . The i -perimeter, or Lusztig parametrization, of b is the multisegment
per i (b) = (1 ,, N),     where    -j βj = μ si1 sij - μ si1 s ij-1 (MVperim)
so that (1 ,, N) is the sequence of lengths μ1 1 μ si1 2 μ si1 si2 3 along the i -perimeter of b. Any perj (b) can be computed from peri (b) by a sequence of "Coxeter relations":
Rij ji (a, a+1) = ( a+1, a) , if si sj = sj si , (Tr1)
Riji jij ( a , a+1 , a+2 ) = ( a+1 + a+2 - min( a , a+2 ) , min( a , a+2 ) , a + a+1 - min( a , a+2 ) ) , if si sj si = sj si sj , (Tr2)

The crystal operator f i1 is given by

peri ( f i1 b) = (1+1 ,, N), if peri (b) = (1 ,, N) , (MVcrystal)
and the i -growth, or string parametrization, of b is
b= f i1 c1 f iN cN b+, where b+= , (MVgrowth)
the polytope which is a single point.

Relating MV-polytopes and column strict tableaux

In type An, the preferred reduced word for w0 is

w0 = s1 s2 s1 s3 s2 s1 sn sn-1 s2 s1 ,
for which the sequence of positive coroots is
ε1- ε2, ε1- ε3, ε1- ε3, ε2- ε3, ε1- ε4, ε2- ε4, ε3- ε4, , ε1- εn+1, ε2- εn+1, , εn- εn+1, .
The MV-polytope determined by
per i (b) = ( 12, 13, 23, ,, nn+1 )
is also given by
b= f 1 c12 f 2 c13 f 1 c23 f 3 c14 f 2 c24 f 1 c34 f n c1n+1 f n-1 c2n+1 f 2 cn-1n+1 f 1 cnn+1 b+,
cij = 1j + 2j + + ij
and b corresponds to the column strict tableau T given by
ij = (number of i in row j of T). (MVtoCST)
(see [MG, Prop. 2.3.13], [Ka] and [BZ]).

Notes and References

This summary of the theory of MV-polyopes is part of joint work with A. Ghitza and S. Kannan on the relationship between MV-cycles and the Borel-Weil-Bott theorem. The theory began from the ideas of [Lusztig????] and Anderson[An], and was developed in Kamnitzer [Km1-2]. The primary references are [Lusztig????], Morier-Genoud [MG],and Kamnitzer [Km1-2].


[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 Ar, 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

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