## MV type ${A}_{2}$

The corresponding dual canonical basis elements are $1 f1 f2 f1∘2 f2f1 f1f2 f2∘2 f1∘3 f1∘f2f1 f1∘f1f2 f2f1∘f2 f1f2∘f2 f2∘3 f1∘4 f1∘2∘f2f1 f1∘2∘f1f2 (f2f1)∘2 f1f2 ∘f2f1 (f1f2)∘2 f2f1 ∘f2∘2 f1f2 ∘f2∘2 f2∘4 f1∘5 f1∘3 ∘f2f1 f1∘3 ∘f1f2 f1∘( f2f1 )∘2 f1∘f1f2∘f2f1 f1∘( f1f2 )∘2 (f2f1 )∘2 ∘f2 f1f2 ∘f2 f1∘f2 (f1f2 )∘2 ∘f2 f2f1 ∘f2∘3 f1f2∘ f2∘3 f2∘5$

## General formulas

If $b={\stackrel{\sim }{f}}_{1}^{{c}_{1}-k}{\stackrel{\sim }{f}}_{2}^{{c}_{2}}{\stackrel{\sim }{f}}_{1}^{{c}_{3}}{b}_{+}then$

 ${e}_{1}=⟨{\alpha }_{1},-{c}_{2}{\alpha }_{2}^{\vee }-{c}_{3}{\alpha }_{3}^{\vee }⟩={c}_{2}-2{c}_{3},\phantom{\rule{2em}{0ex}}{e}_{2}=⟨{\alpha }_{2},-{c}_{3}{\alpha }_{a}^{\vee }⟩={c}_{3},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{e}_{3}=0.$
for the Baumann-Gaussent formula
 ${Z}_{b}=\stackrel{‾}{{y}_{1}\left({t}^{{e}_{1}}ℂ{\left[{t}^{-1}\right]}_{{c}_{1}}^{×}\right){y}_{2}\left({t}^{{e}_{2}}ℂ{\left[{t}^{-1}\right]}_{{c}_{2}}^{×}\right){y}_{1}\left({t}^{{e}_{3}}ℂ{\left[{t}^{-1}\right]}_{{c}_{3}}^{×}\right){t}_{{w}_{0}{\lambda }^{\vee }+{\gamma }^{\vee }}K}$
Then
 $\mathrm{ch}\left({\Delta }_{21}\right)={f}_{2}{f}_{1},\phantom{\rule{2em}{0ex}}\mathrm{ch}\left({\Delta }_{12}\right)={f}_{1}{f}_{2},\phantom{\rule{2em}{0ex}}{f}_{1}\circ {f}_{2}={f}_{1}{f}_{2}+q{f}_{2}{f}_{1},\phantom{\rule{2em}{0ex}}{f}_{2}\circ {f}_{1}={f}_{2}{f}_{1}+q{f}_{1}{f}_{2}.$

If ${\gamma }^{\vee }={\gamma }_{1}{\alpha }_{1}^{\vee }+{\gamma }_{2}{\alpha }_{2}^{\vee }$ with ${\gamma }_{2}\ge {\gamma }_{1}$ then

 ${b}_{k}={\stackrel{\sim }{f}}_{1}^{{\gamma }_{1}-k}{\stackrel{\sim }{f}}_{2}^{{\gamma }_{2}}{\stackrel{\sim }{f}}_{1}^{k}{b}_{+}={\stackrel{\sim }{f}}_{2}^{{\gamma }_{2}-{\gamma }_{1}+k}{\stackrel{\sim }{f}}_{1}^{{\gamma }_{1}}{\stackrel{\sim }{f}}_{1}^{{\gamma }_{1}-k}{b}_{+}$,      for $k=1,2,\dots {\gamma }_{1}$,
with
 ${L}_{{b}_{k}}=\mathrm{hd}\left({\Delta }_{2}^{\circ \left({\gamma }_{2}-{\gamma }_{1}+k\right)}\circ {\Delta }_{12}^{\circ \left({\gamma }_{1}-k\right)}\circ {\Delta }_{1}^{\circ \left(k\right)}\right)=\mathrm{hd}\left({\Delta }_{1}^{\circ \left({\gamma }_{1}-k\right)}\circ {\Delta }_{21}^{\circ \left(k\right)}\circ {\Delta }_{2}^{\circ \left({\gamma }_{2}-k\right)}\right)$,
 ${Z}_{{b}_{k}}=\stackrel{‾}{{y}_{1}\left({t}^{{\gamma }_{2}-2k}ℂ{\left[{t}^{-1}\right]}_{{\gamma }_{1}-k}^{×}\right){y}_{2}\left({t}^{k}ℂ{\left[{t}^{-1}\right]}_{{\gamma }_{2}}^{×}\right){y}_{1}\left(ℂ{\left[{t}^{-1}\right]}_{k}^{×}\right){t}_{{w}_{0}{\lambda }^{\vee }+{\gamma }^{\vee }}K}$
and
 $\mathrm{ch}\left({L}_{{b}_{k}}\right)=\mathrm{ch}\left({Z}_{{b}_{k}}\right)={\left({f}_{1}{f}_{2}\right)}^{\circ k}\circ {\left({f}_{2}{f}_{1}\right)}^{\circ \left({\gamma }_{1}-k\right)}\circ {f}_{2}^{\circ \left({\gamma }_{2}-{\gamma }_{1}\right)}$.
If ${\gamma }^{\vee }={\gamma }_{1}{\alpha }_{1}^{\vee }+{\gamma }_{2}{\alpha }_{2}^{\vee }$ with ${\gamma }_{2}\le {\gamma }_{1}$ then
 ${b}_{k}={\stackrel{\sim }{f}}_{1}^{{\gamma }_{1}-k}{\stackrel{\sim }{f}}_{2}^{{\gamma }_{2}}{\stackrel{\sim }{f}}_{1}^{k}{b}_{+}={\stackrel{\sim }{f}}_{2}^{k}{\stackrel{\sim }{f}}_{1}^{{\gamma }_{1}}{\stackrel{\sim }{f}}_{1}^{{\gamma }_{2}-k}{b}_{+}$,      for $k=1,2,\dots {\gamma }_{2}$,
with
 ${L}_{{b}_{k}}=\mathrm{hd}\left({\Delta }_{2}^{\circ \left(k\right)}\circ {\Delta }_{12}^{\circ \left({\gamma }_{2}-k\right)}\circ {\Delta }_{1}^{\circ \left({\gamma }_{1}-{\gamma }_{2}+k\right)}\right)=\mathrm{hd}\left({\Delta }_{1}^{\circ \left({\gamma }_{1}-k\right)}\circ {\Delta }_{21}^{\circ \left(k\right)}\circ {\Delta }_{2}^{\circ \left({\gamma }_{2}-k\right)}\right)$,
 ${Z}_{{b}_{k}}=\stackrel{‾}{{y}_{1}\left({t}^{{\gamma }_{2}-2k}ℂ{\left[{t}^{-1}\right]}_{{\gamma }_{1}-k}^{×}\right){y}_{2}\left({t}^{k}ℂ{\left[{t}^{-1}\right]}_{{\gamma }_{2}}^{×}\right){y}_{1}\left(ℂ{\left[{t}^{-1}\right]}_{k}^{×}\right){t}_{{w}_{0}{\lambda }^{\vee }+{\gamma }^{\vee }}K}$
and
 $\mathrm{ch}\left({L}_{{b}_{k}}\right)=\mathrm{ch}\left({Z}_{{b}_{k}}\right)={f}_{1}^{\circ \left({\gamma }_{1}-{\gamma }_{2}\right)}\circ {\left({f}_{1}{f}_{2}\right)}^{\circ \left({\gamma }_{2}-k\right)}\circ {\left({f}_{2}{f}_{1}\right)}^{\circ k}$.

Example: If ${\gamma }^{\vee }=5{\alpha }_{1}^{\vee }+7{\alpha }_{2}^{\vee }$ with ${\alpha }_{1}^{\vee }↙$ and ${\alpha }_{2}^{\vee }↘$, then

 ${b}_{3}={\stackrel{\sim }{f}}_{1}^{2}{\stackrel{\sim }{f}}_{2}^{7}{\stackrel{\sim }{f}}_{1}^{3}{b}_{+}={\stackrel{\sim }{f}}_{2}^{5}{\stackrel{\sim }{f}}_{1}^{5}{\stackrel{\sim }{f}}_{2}^{2}{b}_{+}$,
 $k=3$ $γ1-k=2$ $γ2-γ1+k=5$ $2=γ1-k$ $4=γ2-k$ $5=γ1$ $7=γ2$ $3=k$ $2=γ1-k$
the corresponding column strict tableau is
 $b3=$ $33333 ⏟ 2$ $3333333 ⏟ 3$ $3333333333 ⏟ 4$ $1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 3 3 3 2 2 2 2 2 2 3 3 3 3$
and the bead diagram is and
 ${L}_{{b}_{3}}=\mathrm{hd}\left({\Delta }_{2}^{\circ 5}\circ {\Delta }_{12}^{\circ 2}\circ {\Delta }_{1}^{\circ 3}\right)=\mathrm{hd}\left({\Delta }_{1}^{\circ 2}\circ {\Delta }_{21}^{\circ 3}\circ {\Delta }_{2}^{\circ 4}\right)$,
 ${Z}_{{b}_{3}}=\stackrel{‾}{\left\{{y}_{1}\left({a}_{1}{t}^{-1}+{a}_{2}\right){y}_{2}\left({a}_{3}{t}^{-4}+{a}_{4}{t}^{-3}+{a}_{5}{t}^{-2}+{a}_{6}{t}^{-1}+{a}_{7}+{a}_{8}t+{a}_{9}{t}^{2}\right){y}_{1}\left({a}_{10}{t}^{-3}+{a}_{11}{t}^{-2}+{a}_{12}{t}^{-1}\right){t}_{{w}_{0}{\lambda }^{\vee }+{\gamma }^{\vee }}K\phantom{\rule{0.2em}{0ex}}|\phantom{\rule{0.2em}{0ex}}{a}_{i}\in ℂ\right\}}$
and
 $\mathrm{ch}\left({L}_{{b}_{3}}\right)=\mathrm{ch}\left({Z}_{{b}_{3}}\right)={\left({f}_{1}{f}_{2}\right)}^{\circ 2}\circ {\left({f}_{2}{f}_{1}\right)}^{\circ 3}\circ {f}_{2}^{\circ 2}$.
Note that ${\left({f}_{1}{f}_{2}\right)}^{\circ 2}\circ {\left({f}_{2}{f}_{1}\right)}^{\circ 3}\circ {f}_{2}^{\circ 2}={f}_{1}{f}_{1}{f}_{2}{f}_{1}{f}_{2}{f}_{1}{f}_{2}{f}_{1}{f}_{2}{f}_{2}{f}_{2}{f}_{2}+\cdots +{f}_{2}{f}_{2}{f}_{2}{f}_{2}{f}_{2}{f}_{1}{f}_{2}{f}_{1}{f}_{2}{f}_{1}{f}_{1}{f}_{1}$ so that the extremal terms correspond to paths near the boundary of the MV-polytope.

Example: If ${\gamma }^{\vee }=7{\alpha }_{1}^{\vee }+5{\alpha }_{2}^{\vee }$ and $k=3$, with ${\alpha }_{1}^{\vee }↙$ and ${\alpha }_{2}^{\vee }↘$, then

 ${b}_{3}={\stackrel{\sim }{f}}_{1}^{4}{\stackrel{\sim }{f}}_{2}^{5}{\stackrel{\sim }{f}}_{1}^{3}{b}_{+}={\stackrel{\sim }{f}}_{2}^{3}{\stackrel{\sim }{f}}_{1}^{7}{\stackrel{\sim }{f}}_{2}^{2}{b}_{+}$,
 $n1'$ $n2'$ $n3'$ $n1$ $n2$ $n3$ $c1$ $c2$ $c3$ $c1'$ $c2'$ $c3'$
the corresponding column strict tableau is
 $333333333 ⏟ 4$ $3333333 ⏟ 3$ $33333 ⏟ 2$ $1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 3 3 3 2 2 2 2 2 2 2 2 3 3$
and
 ${L}_{{b}_{3}}=\mathrm{hd}\left({\Delta }_{2}^{\circ 3}\circ {\Delta }_{12}^{\circ 2}\circ {\Delta }_{1}^{\circ 5}\right)=\mathrm{hd}\left({\Delta }_{1}^{\circ 4}\circ {\Delta }_{21}^{\circ 3}\circ {\Delta }_{2}^{\circ 2}\right)$,
 ${Z}_{{b}_{3}}=\stackrel{‾}{{y}_{1}\left({t}^{-1}ℂ{\left[{t}^{-1}\right]}_{4}^{×}\right){y}_{2}\left({t}^{3}ℂ{\left[{t}^{-1}\right]}_{5}^{×}\right){y}_{1}\left(ℂ{\left[{t}^{-1}\right]}_{3}^{×}\right){t}_{{w}_{0}{\lambda }^{\vee }+{\gamma }^{\vee }}K}$
and
 $\mathrm{ch}\left({L}_{{b}_{3}}\right)=\mathrm{ch}\left({Z}_{{b}_{3}}\right)={f}_{1}^{\circ 2}\circ {\left({f}_{1}{f}_{2}\right)}^{\circ 2}\circ {\left({f}_{2}{f}_{1}\right)}^{\circ 3}$.

## Notes and References

This analysis of MV-cycles and irreducible characters of quiver Hecke algebras for type ${A}_{2}$ 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 $A2$ 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 ${A}_{r}$, 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