Math 610 Assignment 2

1. Let ${\bf i} = (1,2,1)$. Check that $T_1 T_2 F_1 = F_2$ using the defining relations of $U_q(\mathfrak{sl}_3)$ as in Tingley.
2. Fix $w_0 = s_1 s_2 s_1$ in $W = S_3$ and use $W$-invariance of the bilinear form on the dual of the Cartan subalgebra $\mathfrak{t}$ of $\mathfrak{sl}_3$ to check that $\langle \alpha_1 , w_0 \omega_2^\vee\rangle = -1$. Here $\alpha_i$ denotes a simple root and $\omega_i^\vee$ denotes its coroot.
3. Let $P = \text{Hom}(T,\mathbb C^\times)$ and $P^\vee = \text{Hom}(\mathbb C^\times, T)$ where $T\le G$ denotes the subgroup of diagonal matrices. Describe $P, P^\vee$ in each case below, and state any generalizations you foresee. All homs are algebraic homs, and you may use as definition that $\text{Hom}(\mathbb C^\times,\mathbb C^\times) = \text{Hom}(\mathbb C[z,z^{-1}],\mathbb C[z,z^{-1}])= \{z\mapsto z^n : n\in \mathbb Z \}$. In particular, an algebraic hom is determined by its derivative at the identity ($1 \in \mathbb C^\times$). This allows us to identify $P$ in $\mathfrak t^\ast = (T_e T)^\ast = \text{Hom}(T_eT,\mathbb C)=T_e\text{Hom}(T,\mathbb C^\times)$ for example.
   • $G = \text{GL}_2$
   • $G = \text{GL}_3$
   • $G = \text{PGL}_2$ where $\text{PGL}_{n+1} := \left(\text{GL}_{n+1} \setminus \{0\}\right)/{\mathbb C}^\times$ with $\lambda\in\mathbb C^\times$ acting on $g\in\text{GL}_{n+1}$ by scaling.
   • $G = \text{PGL}_3$
   • $G = \text{SL}_2$ where $\text{SL}_{n+1} := \left\{g\in\text{GL}_{n+1}: \det g = 1\right\}$
   • $G = \text{SL}_3$
4. For each of the above, describe the root basis (resp. coroot basis) of $P$ (resp. $P^\vee$) in $\mathfrak t^\ast$ (resp. $\mathfrak t$). (Start by recalling the definitions-how are co/roots defined?)
5. Repeat the last question, with "root" replaced by "weight". (Start by recalling the definitions-how are co/weights defined?)
6. Let $\Gamma = \{w\omega_i^\vee : w\in W, i\in I\}$. Recall that a BZ datum $M = (M_\gamma)_\Gamma$ of weight $(\lambda,\mu)$ is a $\Gamma$-tuple of integers satisfying (i) the edge inequalities, (ii) the tropical Plucker relations, and (iii) $\langle w_0\omega_i^\vee,\lambda\rangle = M_{w_0\omega_i^\vee}$ and $\langle \omega_i^\vee,\mu\rangle = M_{\omega_i^\vee}$ for all $i\in I$. The associated MV polytope $P(M) = \{\nu \in \mathfrak{t}^\ast : \langle \nu , \gamma\rangle \ge M_\gamma \}$ is a convex polytope (with lowest weight $\mu$ and highest weight $\lambda$). Given $w\in W$, let $\mu_w\in P$ be such that $\langle \mu_w , w\omega_i^\vee\rangle = M_{w\omega_i}$. Show that $P(M)$ is the convex hull of $\{\mu_w : w\in W\}$.
7. Let $G = \text{SL}_3$.
   • Find the possible $P(M)$ when $M$ is a BZ datum of weight $(\alpha_1,0)$.
   • Find the possible $P(M)$ when $M$ is a BZ datum of weight $(\alpha_1 + \alpha_2, \alpha_1)$.
   • Describe the possible $P(M)$ for a general BZ datum $M$.
8. Prove that $y\mapsto [w_0 y^T]_+$ is a birational automorphism of $U$ with inverse $x\mapsto w_0^{-1}[xw_0^{-1}]_+^Tw_0$.
9. Let $Q$ be the quiver with just one vertex $1$ and arrows $a,b$. Set $A = \mathbb C Q/ J$ where $J$ is the ideal generated by $\{ab,ba\}$. Describe the modules $V_{\bf i}^k$ defined in class for ${\bf i} = (i_4,\dots,i_1) = (1,1,1,1)$. See section 3.4 of Generic bases for cluster algebras and the chamber ansatz by Geiss, Leclerc, Schroer for a reminder.

References

• Zelevinsky
• Kamnitzer
• Gelfand-Zelevinsky
• Berenstein-Zelevinsky
• Berenstein-Kazhdan
• Assem-Simson-Skowronski: Elements of the representation theory of associative algebras
• Geiss-Leclerc-Schroer: Generic bases for cluster algebras and the chamber ansatz
• Serre: Complex semisimple Lie algebras