Degree bounds for semi-invariant rings of quivers

Abstract

Abstract. We use recent results on matrix semi-invariants to give degree bounds on generators for the ring of semi-invariants for quivers with no oriented cycles.

Generating invariant rings of quivers in arbitrary characteristic

“…We showed the above results in characteristic 0 in [5,6]. In this paper, we show that the restrictions on characteristic can be removed.…”

“…The invariant rings under consideration are CohenMacaulay by Corollary 4.3, and that the degree of the Hilbert series is independent of the underlying field by Proposition 5.1 and Corollary 5.2. Also note that the results on the null cone in [5,6] were independent of the underlying field K. Hence the arguments in [5,6] Observe that a set of generating invariants is a separating subset, and hence we have…”

Non-commutative Rank and Semi-stability of Quiver Representations

“…We may also write ncrk(A) instead of ncrk(A). The value of (rk A {d} )/d is increasing as d increases, and is bounded by n. Derksen and Makam proved that if A has maximal noncommutative rank, then taking d ≥ n−1 ensures rk A {d} = nd [DM18a]. If ncrk(A) = r < n, then restricting to a full rank r × r submatrix of A(x), we see that rk A {d} = nd for d ≥ r − 1.…”

“…To do this, we reduce any acyclic quiver to the Kronecker quiver. We use the construction described in [DM18a], but provide an altered set up, using presentations as in [DF15]. Let P x be the indecomposable representation of Q with basis given by all paths starting at vertex x.…”

“…This randomized algorithm is much simpler and typically must faster than the deterministic algorithm. In the context of representations, this algorithm immediately after reduction, blow up by a sufficiently large [IQS18,DM18a] factor, d ≥ n − 1. In this blow-up, randomly choose a matrix A :…”