## MV polytopes

A MV-polytope is $b=\mathrm{Conv}\left\{{\mu }_{w}|w\in W\right\}$ 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
 ${\beta }_{1}^{\vee }={\alpha }_{{i}_{1}}^{\vee },\phantom{\rule{.5em}{0ex}}{\beta }_{2}^{\vee }={s}_{{i}_{1}}{\alpha }_{{i}_{2}}^{\vee },\phantom{\rule{.5em}{0ex}}{\beta }_{N}^{\vee }={s}_{{i}_{1}}\cdots {s}_{{i}_{N-1}}{\alpha }_{{i}_{N}}^{\vee }$. (rootorder)
A multisegment is a sequence $\left({\ell }_{1},\dots ,{\ell }_{N}\right)\in {\left({ℤ}_{\ge 0}\right)}^{N}$. The $\stackrel{\to }{i}$-perimeter, or Lusztig parametrization, of $b$ is the multisegment
 ${\mathrm{per}}_{\stackrel{\to }{i}}\left(b\right)=\left({\ell }_{1},\dots ,{\ell }_{N}\right)$,     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 $\left({\ell }_{1},\dots ,{\ell }_{N}\right)$ 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 $\stackrel{\to }{i}$-perimeter of $b$. Any ${\mathrm{per}}_{\stackrel{\to }{j}}\left(b\right)$ can be computed from ${\mathrm{per}}_{\stackrel{\to }{i}}\left(b\right)$ by a sequence of "Coxeter relations":
 $R_{ij}^{ji}\left({\ell }_{a},{\ell }_{a+1}\right)=\left({\ell }_{a+1},{\ell }_{a}\right)$, if ${s}_{i}{s}_{j}={s}_{j}{s}_{i}$, (Tr1)
 $R_{iji}^{jij}\left({\ell }_{a},{\ell }_{a+1},{\ell }_{a+2}\right)=\left({\ell }_{a+1}+{\ell }_{a+2}-\mathrm{min}\left({\ell }_{a},{\ell }_{a+2}\right),\mathrm{min}\left({\ell }_{a},{\ell }_{a+2}\right),{\ell }_{a}+{\ell }_{a+1}-\mathrm{min}\left({\ell }_{a},{\ell }_{a+2}\right)\right)$, 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

 ${\mathrm{per}}_{\stackrel{\to }{i}}\left({\stackrel{\sim }{f}}_{{i}_{1}}b\right)=\left({\ell }_{1}+1,\dots ,{\ell }_{N}\right),\phantom{\rule{.5em}{0ex}}\text{if}\phantom{\rule{.5em}{0ex}}{\mathrm{per}}_{\stackrel{\to }{i}}\left(b\right)=\left({\ell }_{1},\dots ,{\ell }_{N}\right),$ (MVcrystal)
and the $\stackrel{\to }{i}$-growth, or string parametrization, of $b$ is
 $b={\stackrel{\sim }{f}}_{{i}_{1}}^{{c}_{1}}\cdots {\stackrel{\sim }{f}}_{{i}_{N}}^{{c}_{N}}{b}_{+},\phantom{\rule{.5em}{0ex}}\text{where}\phantom{\rule{.5em}{0ex}}{b}_{+}=•$, (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
 ${\epsilon }_{1}-{\epsilon }_{2},\phantom{\rule{.2em}{0ex}}{\epsilon }_{1}-{\epsilon }_{3},\phantom{\rule{.2em}{0ex}}{\epsilon }_{1}-{\epsilon }_{3},\phantom{\rule{.2em}{0ex}}{\epsilon }_{2}-{\epsilon }_{3},\phantom{\rule{.2em}{0ex}}{\epsilon }_{1}-{\epsilon }_{4},\phantom{\rule{.2em}{0ex}}{\epsilon }_{2}-{\epsilon }_{4},\phantom{\rule{.2em}{0ex}}{\epsilon }_{3}-{\epsilon }_{4},\phantom{\rule{.2em}{0ex}}\dots \phantom{\rule{.1em}{0ex}},\phantom{\rule{.2em}{0ex}}{\epsilon }_{1}-{\epsilon }_{n+1},\phantom{\rule{.2em}{0ex}}{\epsilon }_{2}-{\epsilon }_{n+1},\phantom{\rule{.2em}{0ex}}\dots \phantom{\rule{.1em}{0ex}},\phantom{\rule{.2em}{0ex}}{\epsilon }_{n}-{\epsilon }_{n+1},$.
The MV-polytope determined by
 ${\mathrm{per}}_{\stackrel{\to }{i}}\left(b\right)=\left({\ell }_{\mathrm{12}},{\ell }_{\mathrm{13}},{\ell }_{\mathrm{23}},,\dots ,{\ell }_{nn+1}\right)$
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
 ${c}_{\mathrm{ij}}={\ell }_{\mathrm{1j}}+{\ell }_{\mathrm{2j}}+\cdots +{\ell }_{\mathrm{ij}}$
and $b$ corresponds to the column strict tableau $T$ given by
 ${\ell }_{\mathrm{ij}}=$ (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].

## 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