# Invariant ring of Aut(V,\xi)

### Fawad Hussain

Friday 28th January, 2011 16:00-17:00 Mathematics Building, room 516

#### Abstract

Let V be a vector space over a field F_q. Suppose e_1,...,e_n is a basis of V. Define G = Aut(V,\xi)=\{g \in GL(V) : \xi(gu,gv) = \xi(u,v) for all u,v \in V \} Let S = F_q[x_1,...,x_n] where x_1,...,x_n is the dual basis of V* and x_i(e_j)= \delta_{ij} where \delta_{ij} = 1 if i = j and \delta_{ij} = 0 if i\neq 0. G acts on V* as follows x_i^g(e_j) = x_i(ge_j) Action of G on V* extends to an action on S by ring automorphism ie (x + y)^g = x^g + y^g, (xy)^g = x^g y^g and 1^g = 1 for all x, y \in S. In this talk i will discuss the invariant ring S^G = \{f \in S : f^g = f for all g \in G}.