MV polytopes

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 10 February 2011

MV polytopes

A MV-polytope is $b=\mathrm{Conv}\left\{{\mu }_w\right|w\in W\}$ the convex hull of its vertices.

${\mu }_1$ is the type of $b$,     and     ${\mu }_{w_0}$ is the weight of $b$.

A reduced word $w_0=s_{i_1}\cdots s_{i_N}$ for $w_0$ induces an ordering on the positive coroots

${\beta }_1^{\vee }={\alpha }_{i_1}^{\vee },{\beta }_2^{\vee }=s_{i_1}{\alpha }_{i_2}^{\vee },{\beta }_N^{\vee }=s_{i_1}\cdots s_{i_{N-1}}{\alpha }_{i_{\mathrm{N}}}^{\vee }$.

(rootorder)

A multisegment is a sequence $({\ell }_1,\dots ,{\ell }_N)\in {\left({\mathbb{Z}}_{\ge 0}\right)}^N$. The $\overrightarrow{i}$-perimeter, or Lusztig parametrization, of $b$ is the multisegment

${\mathrm{per}}_{\overrightarrow{i}}\left(b\right)=({\ell }_1,\dots ,{\ell }_N)$,     where    $-{\ell }_j{\beta }_j^{\vee }={\mu }_{s_{i_1}\cdots s_{i_j}}-{\mu }_{s_{i_1}\cdots s_{i_{j-1}}}$

(MVperim)

so that $({\ell }_1,\dots ,{\ell }_N)$ is the sequence of lengths ${\mu }_1\stackrel{{\ell }_1}{\to }{\mu }_{s_{i_1}}\stackrel{{\ell }_2}{\to }{\mu }_{s_{i_1}{s}_{i_2}}\stackrel{{\ell }_3}{\to }\cdots$ along the $\overrightarrow{i}$-perimeter of $b$. Any ${\mathrm{per}}_{\overrightarrow{j}}\left(b\right)$ can be computed from ${\mathrm{per}}_{\overrightarrow{i}}\left(b\right)$ by a sequence of "Coxeter relations":

$R_{ij}^{ji}({\ell }_a,{\ell }_{a+1})=({\ell }_{a+1},{\ell }_a)$, if $s_i{s}_j=s_j{s}_i$,

(Tr1)

$R_{iji}^{jij}({\ell }_a,{\ell }_{a+1},{\ell }_{a+2})=({\ell }_{a+1}+{\ell }_{a+2}-\min ({\ell }_a,{\ell }_{a+2}),\min ({\ell }_a,{\ell }_{a+2}),{\ell }_a+{\ell }_{a+1}-\min ({\ell }_a,{\ell }_{a+2}\left)\right)$, if $s_i{s}_j{s}_i=s_j{s}_i{s}_j$,

(Tr2)

The crystal operator ${\stackrel{\sim }{f}}_{i_1}$ is given by

${\mathrm{per}}_{\overrightarrow{i}}\left({\stackrel{\sim }{f}}_{i_1}b\right)=({\ell }_1+1,\dots ,{\ell }_N),\text{if}$ {\mathrm{per}}_{\overrightarrow{i}}\left(b\right)=({\ell }_1,\dots ,{\ell }_N)$,$

(MVcrystal)

and the $\overrightarrow{i}$-growth, or string parametrization, of $b$ is

$b={\stackrel{\sim }{f}}_{i_1}^{c_1}\cdots {\stackrel{\sim }{f}}_{i_N}^{c_N}{b}_{+},\text{where}{b}_{+}=•$,

(MVgrowth)

the polytope which is a single point.

Relating MV-polytopes and column strict tableaux

In type $A_n$, the preferred reduced word for $w_0$ is

$w_0=s_1{s}_2{s}_1{s}_3{s}_2{s}_1\cdots s_n{s}_{n-1}\cdots s_2{s}_1$,

for which the sequence of positive coroots is

${\epsilon }_1-{\epsilon }_2,{\epsilon }_1-{\epsilon }_3,{\epsilon }_1-{\epsilon }_3,{\epsilon }_2-{\epsilon }_3,{\epsilon }_1-{\epsilon }_4,{\epsilon }_2-{\epsilon }_4,{\epsilon }_3-{\epsilon }_4,\dots ,{\epsilon }_1-{\epsilon }_{n+1},{\epsilon }_2-{\epsilon }_{n+1},\dots ,{\epsilon }_n-{\epsilon }_{n+1},$.

The MV-polytope determined by

${\mathrm{per}}_{\overrightarrow{i}}\left(b\right)=({\ell }_{\mathrm{<mn>1</mn><mn>2</mn>}},{\ell }_{\mathrm{<mn>1</mn><mn>3</mn>}},{\ell }_{\mathrm{<mn>2</mn><mn>3</mn>}},,\dots ,{\ell }_{nn+1})$

is also given by

$b={\stackrel{\sim }{f}}_1^{c_{12}}{\stackrel{\sim }{f}}_2^{c_{13}}{\stackrel{\sim }{f}}_1^{c_{23}}{\stackrel{\sim }{f}}_3^{c_{14}}{\stackrel{\sim }{f}}_2^{c_{24}}{\stackrel{\sim }{f}}_1^{c_{34}}\cdots {\stackrel{\sim }{f}}_n^{c_{1n+1}}{\stackrel{\sim }{f}}_{n-1}^{c_{2n+1}}\cdots {\stackrel{\sim }{f}}_2^{c_{n-1n+1}}{\stackrel{\sim }{f}}_1^{c_{nn+1}}{b}_{+},$

where

and $b$ corresponds to the column strict tableau $T$ given by

(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].

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 $A_r$, 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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