MV type $A_2$

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 $$\begin{array}{ccccccccccc}& & & & & 1& & & & & \\ & & & & f_1& & f_2& & & & \\ & & & f_1^{\circ 2}& & \genfrac{}{}{0ex}{}{f_2{f}_1}{f_1{f}_2}& & f_2^{\circ 2}& & & \\ & & f_1^{\circ 3}& & \genfrac{}{}{0ex}{}{f_1\circ f_2{f}_1}{f_1\circ f_1{f}_2}& & \genfrac{}{}{0ex}{}{f_2{f}_1\circ f_2}{f_1{f}_2\circ f_2}& & f_2^{\circ 3}& & \\ & f_1^{\circ 4}& & \genfrac{}{}{0ex}{}{f_1^{\circ 2}\circ f_2{f}_1}{f_1^{\circ 2}\circ f_1{f}_2}& & \genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{(f_2{f}_1{)}^{\circ 2}}{f_1{f}_2\circ f_2{f}_1}}{(f_1{f}_2{)}^{\circ 2}}& & \genfrac{}{}{0ex}{}{f_2{f}_1\circ f_2^{\circ 2}}{f_1{f}_2\circ f_2^{\circ 2}}& & f_2^{\circ 4}& \\ f_1^{\circ 5}& & \genfrac{}{}{0ex}{}{f_1^{\circ 3}\circ f_2{f}_1}{f_1^{\circ 3}\circ f_1{f}_2}& & \genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{f_1\circ (f_2{f}_1{)}^{\circ 2}}{f_1\circ f_1{f}_2\circ f_2{f}_1}}{f_1\circ (f_1{f}_2{)}^{\circ 2}}& & \genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{(f_2{f}_1{)}^{\circ 2}\circ f_2}{f_1{f}_2\circ f_2{f}_1\circ f_2}}{(f_1{f}_2{)}^{\circ 2}\circ f_2}& & \genfrac{}{}{0ex}{}{f_2{f}_1\circ f_2^{\circ 3}}{f_1{f}_2\circ f_2^{\circ 3}}& & f_2^{\circ 5}\end{array}$$

General formulas

If $b={\stackrel{\sim }{f}}_1^{c_1-k}{\stackrel{\sim }{f}}_2^{c_2}{\stackrel{\sim }{f}}_1^{c_3}{b}_{+}then$

$e_1=⟨{\alpha }_1,-c_2{\alpha }_2^{\vee }-c_3{\alpha }_3^{\vee }⟩=c_2-2c_3,e_2=⟨{\alpha }_2,-c_3{\alpha }_{\mathrm{a}}^{\vee }⟩=c_3,\text{and}{e}_3=0.$

for the Baumann-Gaussent formula

$Z_b=\overline{y_1\left(t^{e_1}ℂ{\left[t^{-1}\right]}_{c_1}^{\times }\right)y_2\left(t^{e_2}ℂ{\left[t^{-1}\right]}_{c_2}^{\times }\right)y_1\left(t^{e_3}ℂ{\left[t^{-1}\right]}_{c_3}^{\times }\right)t_{w_0{\lambda }^{\vee }+{\gamma }^{\vee }}K}$

Then

$\mathrm{ch}\left({\Delta }_{21}\right)=f_2{f}_1,\mathrm{ch}\left({\Delta }_{12}\right)=f_1{f}_2,f_1\circ f_2=f_1{f}_2+q{f}_2{f}_1,f_2\circ f_1=f_2{f}_1+q{f}_1{f}_2.$

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

$L_{b_k}=\mathrm{hd}({\Delta }_2^{\circ ({\gamma }_2-{\gamma }_1+k)}\circ {\Delta }_{12}^{\circ ({\gamma }_1-k)}\circ {\Delta }_1^{\circ \left(k\right)})=\mathrm{hd}({\Delta }_1^{\circ ({\gamma }_1-k)}\circ {\Delta }_{21}^{\circ \left(k\right)}\circ {\Delta }_2^{\circ ({\gamma }_2-k)})$,

$Z_{b_k}=\overline{y_1\left(t^{{\gamma }_2-2k}ℂ{\left[t^{-1}\right]}_{{\gamma }_1-k}^{\times }\right)y_2\left(t^kℂ{\left[t^{-1}\right]}_{{\gamma }_2}^{\times }\right)y_1\left(ℂ{\left[t^{-1}\right]}_k^{\times }\right)t_{w_0{\lambda }^{\vee }+{\gamma }^{\vee }}K}$

and

$\mathrm{ch}\left(L_{b_k}\right)=\mathrm{ch}\left(Z_{b_k}\right)={\left(f_1{f}_2\right)}^{\circ k}\circ {\left(f_2{f}_1\right)}^{\circ ({\gamma }_1-k)}\circ f_2^{\circ ({\gamma }_2-{\gamma }_1)}$.

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

$L_{b_k}=\mathrm{hd}({\Delta }_2^{\circ \left(k\right)}\circ {\Delta }_{12}^{\circ ({\gamma }_2-k)}\circ {\Delta }_1^{\circ ({\gamma }_1-{\gamma }_2+k)})=\mathrm{hd}({\Delta }_1^{\circ ({\gamma }_1-k)}\circ {\Delta }_{21}^{\circ \left(k\right)}\circ {\Delta }_2^{\circ ({\gamma }_2-k)})$,

$Z_{b_k}=\overline{y_1\left(t^{{\gamma }_2-2k}ℂ{\left[t^{-1}\right]}_{{\gamma }_1-k}^{\times }\right)y_2\left(t^kℂ{\left[t^{-1}\right]}_{{\gamma }_2}^{\times }\right)y_1\left(ℂ{\left[t^{-1}\right]}_k^{\times }\right)t_{w_0{\lambda }^{\vee }+{\gamma }^{\vee }}K}$

and

$\mathrm{ch}\left(L_{b_k}\right)=\mathrm{ch}\left(Z_{b_k}\right)=f_1^{\circ ({\gamma }_1-{\gamma }_2)}\circ {\left(f_1{f}_2\right)}^{\circ ({\gamma }_2-k)}\circ {\left(f_2{f}_1\right)}^{\circ k}$.

Example: If ${\gamma }^{\vee }=5{\alpha }_1^{\vee }+7{\alpha }_2^{\vee }$ with ${\alpha }_1^{\vee }\swarrow$ and ${\alpha }_2^{\vee }\searrow$, then

$b_3={\stackrel{\sim }{f}}_1^2{\stackrel{\sim }{f}}_2^7{\stackrel{\sim }{f}}_1^3{b}_{+}={\stackrel{\sim }{f}}_2^5{\stackrel{\sim }{f}}_1^5{\stackrel{\sim }{f}}_2^2{b}_{+}$,

the corresponding column strict tableau is

and the bead diagram is and

$L_{b_3}=\mathrm{hd}({\Delta }_2^{\circ 5}\circ {\Delta }_{12}^{\circ 2}\circ {\Delta }_1^{\circ 3})=\mathrm{hd}({\Delta }_1^{\circ 2}\circ {\Delta }_{21}^{\circ 3}\circ {\Delta }_2^{\circ 4})$,

$Z_{b_3}=\stackrel{‾}{\left\{y_1(a_1{t}^{-1}+a_2)y_2(a_3{t}^{-4}+a_4{t}^{-3}+a_5{t}^{-2}+a_6{t}^{-1}+a_7+a_{8}t+a_9{t}^2)y_1(a_{10}{t}^{-3}+a_{11}{t}^{-2}+a_{12}{t}^{-1})t_{w_0{\lambda }^{\vee }+{\gamma }^{\vee }}K\right|a_i\in ℂ\}}$

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 ${\alpha }_1^{\vee }\swarrow$ and ${\alpha }_2^{\vee }\searrow$, then

$b_3={\stackrel{\sim }{f}}_1^4{\stackrel{\sim }{f}}_2^5{\stackrel{\sim }{f}}_1^3{b}_{+}={\stackrel{\sim }{f}}_2^3{\stackrel{\sim }{f}}_1^7{\stackrel{\sim }{f}}_2^2{b}_{+}$,

the corresponding column strict tableau is

and the bead diagram is

PICTURE OF BEAD DIAGRAM HERE

and

$L_{b_3}=\mathrm{hd}({\Delta }_2^{\circ 3}\circ {\Delta }_{12}^{\circ 2}\circ {\Delta }_1^{\circ 5})=\mathrm{hd}({\Delta }_1^{\circ 4}\circ {\Delta }_{21}^{\circ 3}\circ {\Delta }_2^{\circ 2})$,

$Z_{b_3}=\overline{y_1\left(t^{-1}ℂ{\left[t^{-1}\right]}_4^{\times }\right)y_2\left(t^3ℂ{\left[t^{-1}\right]}_5^{\times }\right)y_1\left(ℂ{\left[t^{-1}\right]}_3^{\times }\right)t_{w_0{\lambda }^{\vee }+{\gamma }^{\vee }}K}$

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

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