2.00 pm, Seminar Hall
Invariants of several matrices under SL(n) x SL(n) action
K V Subrahmanyam
Chennai Mathematical Institute.
Given m, n x n matrices (X_1,X_2,....,X_m) with entries in a field F, the group SL(n,F) x SL(n,F) acts on this m-tuple with (A,B) sending the m tuple to (A X_1 B^t, AX_2B^t,... ,AX_mB^t). A description of the polynomial functions which are invariant under this action is well known (over infinite fields). Over infinite fields a description of the null cone is also known, but bounds on the degree in which the invariant ring is generated is not known. Recently, based on our result on the rank of matrix families under blow-ups, Derksen and Vishwambara showed that the invariant ring is generated in degree n^6 (over infinite fields)
I will describe our result, regularity under blow-ups, and the Derksen and Vishwambara result. I will also indicate how a simple calculation starting with our result yields the same n^6 bound, without using the Derksen Vishwambara machinery.
I will also show that testing membership in the null cone --do all invariant functions vanish on a given tuple of matrices (C_1,C_2,...,C_m)--, is in polynomial time, improving on a result of Ankit Garg, et al. This also settles the problem of computing in polynomial time the non commutative rank of a family of matrices.
This is joint work with Gabor Ivanyos and Jimmy Qiao.