K V Subrahmanyam: Invariants of several matrices under SL(n) x SL(n) action
Автор: WACT 2016
Загружено: 2016-03-16
Просмотров: 236
Given m, n x n matrices (X1,X2,....,Xm) 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 X1 Bt, AX2Bt,... ,AXmBt). A description of the polynomial functions which are invariant under this action is well known (over infinite fields). This ring of invariant functions is known to be finitely generated. However degree bounds are poor.
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 n6 (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 (almost) the same bounds, 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 (C1,C2,...,Cm) —, 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.
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