b-Functions associated with quivers of type A

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

“…Among the new results in this paper are the "diagonal-parametrized" Cayley identities (Theorems 2.10 and 2.11), the "Laplacian-parametrized" Cayley identities (Theorems 2.12 and 2.14), and the "product-parametrized" and "borderparametrized" rectangular Cayley identities (Theorems 2.16 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.…”

“…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.…”

Singularity invariants related to Milnor fibers: survey

“…The decomposition formula for b-functions, which asserts that under certain conditions, the b-functions of reducible pvs have decompositions correlated to the decomposition of representations, was given in [113]. By using the decomposition formula, the b-functions of relative invariants arising from the quivers of type A have been determined in [119].…”

“…Example. Some interesting examples of b-functions, related to quivers of type A and to generic determinantal varieties with blocks of zeros inserted, are computed in [119]. For example, let X, Y, Z be matrices of three distinct sets of indeterminates of sizes (n 2 , n 1 ), (n 2 , n 3 ), (n 4 , n 3 ), respectively, such that n 1 + n 3 = n 2 + n 4 and n 1 < n 2 .…”