b-Functions associated with quivers of type A

Abstract

We study the b-functions of relative invariants of the prehomogeneous vector spaces associated with quivers of type A. By applying the decomposition formula for b-functions, we determine explicitly the b-functions of one variable for each irreducible relative invariant. Moreover, we give a graphical algorithm to determine the b-functions of several variables.

Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians

“…Finally, the most difficult result obtained in the present paper -namely, the multi-matrix rectangular Cayley identity identity (2.23) -has very recently been proven independently by Sugiyama [100, Theorem 0.1] in the context of prehomogeneous vector spaces associated to equioriented quivers of type A; his proof uses a decomposition formula found earlier by himself and F. Sato [85]. Indeed, Sugiyama proved an even more general result [100,Theorem 3.4], applying to quivers of type A with arbitrary orientation. 17 It is worth stressing that the Cayley identity (1.1) -though not, as far as we can tell, the all-minors version (1.2) -is an immediate consequence of a deeper identity due to Capelli [16][17][18], in which the operator H = (det X)(det ∂) is represented as a noncommutative determinant involving the gl(n) generators X T ∂: see e.g.…”

“…16 and 2.17). We also give an elementary (though rather intricate) proof of the multi-matrix rectangular Cayley identity (Theorem 2.9) that was proven recently by Sugiyama [100].…”

“…Many of the polynomials P treated here can also be understood as relative invariants of prehomogeneous vector spaces [52,60]; and the corresponding b-functions have been computed in that context, mostly by means of microlocal calculus [59,86,100]. See [60,Appendix] for a table of all irreducible reduced prehomogeneous vector spaces and their relative invariants and b-functions.…”

“…for an m × n rectangular matrix (symmetric) s(s + 2) · · · (s + 2m − 2)(s + 2n − 2m + 1) · · · (s + 2n − 1) for an m × n rectangular matrix (antisymmetric) ℓ α=1 n 1 −1 j=0 (s + n α − n 1 + j) for ℓ matrices of sizes n α × n α+1 (9.2) Indeed, the polynomials P occurring in these identities all correspond to relative invariants of prehomogeneous vector spaces: the ordinary, symmetric and antisymmetric Cayley identities [(2.1), (2.4) and (2.7)] correspond to cases (1), (2) and (3), respectively, in Kimura's [60,Appendix] (15) and (13) in the same table; while the two-matrix and multi-matrix rectangular Cayley identities [(2.16) and (2.23)] correspond to prehomogeneous vector spaces associated to equioriented quivers of type A [100]. Whenever P is a relative invariant of a prehomogeneous vector space, the general theory [52,60] allows the immediate identification of a suitable operator Q(∂/∂x) -namely, the dual of P itself -and provides a proof that the corresponding b(s) satisfies deg b = deg P and is indeed (up to a constant factor) the Bernstein-Sato polynomial of P .…”

“…When applicable, this connection allows the immediate identification of a suitable operator Q(∂/∂x) -namely, the dual of P itself -and provides a general proof that the corresponding b(s) satisfies deg b = deg P and is indeed (up to a constant factor) the Bernstein-Sato polynomial of P . 6 Furthermore, this approach sometimes allows the explicit calculation of b(s) by means of microlocal calculus [59,77,78,85,86,99,100,106,109] or other methods [98]. 7 The purpose of the present paper is to give straightforward (and we hope elegant) algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new.…”