Given a permutation w ∈ Sn , we consider a determinantal ideal Iw whose generators are certain minors in the generic n × n matrix (ﬁlled with independent variables). Using ‘multidegrees’ as simple algebraic substitutes for torus-equivariant cohomology classes on vector spaces, our main theorems describe, for each ideal Iw : • variously graded multidegrees and Hilbert series in terms of ordinary and double Schubert and Grothendieck polynomials;
This volume contains the Proceedings of the Workshop "Physics and
Combinatorics" held at the Graduate School of Mathematics, Nagoya University,
Japan, during August 21-26, 2000. The workshop organizing committee
consisted of Kazuhiko Aomoto, Fumiyasu Hirashita, Anatol Kirillov, Ryoichi
Kobayashi, Akihiro Tsuchiya, and Hiroshi Umemura.
We consider a specialization of an untwisted quantum aﬃne algebra of type ADE at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its ﬁnite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of “computable” polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we “compute” q-characters for all simple modules. The result is based on “computations” of Betti numbers of graded/cyclic quiver varieties.