Carol Chang - Representations of quivers with free modules of covariants

Next: Adriano Garsia - An Up: Algebraic Combinatorics, Group Representations Previous: François Bergeron - Diagonal

Carol Chang - Representations of quivers with free modules of covariants

CAROL CHANG, Department of Mathematics, Northeastern University, Boston, Massachusetts 02115, USA

Representations of quivers with free modules of covariants

A quiver is an oriented graph Q=(Q₀,Q₁) where Q₀ is the set of vertices and Q₁ is the set of arrows. For $\alpha \in Q_1$ , $\alpha\colon t\alpha \rightarrow h\alpha$ . A representation V of a quiver Q is a collection $V =\bigl\{\bigl(V_x,V(\alpha)\bigr)\mid x\in Q_0, \alpha \in Q_1\bigr\}$ where V_x is a vector space and $V(\alpha)$ is a linear map from $V_{t\alpha}$ to $V_{h\alpha}$ . Specifiying a dimension at each vertex of the quiver, a representation is then determined by a point of the affine space $\textrm{Rep}(Q,{\bf d}) = \bigoplus_{\alpha\in Q_1}\textrm{Hom}_k(V_{t\alpha},V_{h\alpha})$ . There is a natural action of $\textrm{SL}(Q,{\bf d}) = \prod_{x\in Q_0}\textrm{SL}_{d(x)}(k)$ on $\textrm{Rep}(Q,{\bf d})$ .

Given a finite connected quiver Q, we are interested in when the action of $\textrm{SL}(Q,{\bf d})$ on $\textrm{Rep}(Q,{\bf d})$ gives a cofree representation. In particular, we are interested in studying the situation when the modules of covariants are free $k[\textrm{Rep}(Q,{\bf d})]^{\textrm{SL}(Q,{\bf d})}$ -modules. We will discuss when quivers have free modules of covariants. We will also discuss the combinatorics involved in describing the orbits of the group action mentioned above.

Next: Adriano Garsia - An Up: Algebraic Combinatorics, Group Representations Previous: François Bergeron - Diagonal