The direct product of two Hilbert schemes of the same surface has natural K-theory classes given by the alternating Ext groups between the two ideal sheaves in question, twisted by a line bundle. We express the Chern classes of these virtual bundles in terms of Nakajima operators.
We prove a Littlewood-Richardson type formula for $(s_{lambda/mu},s_{ u/kappa})_{t^k,t}$, the pairing of two skew Schur functions in the MacDonald inner product at $q = t^k$ for positive integers $k$. This pairing counts graded decomposition numbers
in the representation theory of wreath products of the algebra $C[x]/x^k$ and symmetric groups.
We study supersymmetric gauge theories in five dimensions, using their relation to the K-theory of the moduli spaces of torsion free sheaves. In the spirit of the BPS/CFT correspondence the partition function and the expectation values of the chiral,
BPS protected observables are given by the matrix elements and more generally by the correlation functions in some q-deformed conformal field theory in two dimensions. We show that the coupling of the gauge theory to the bi-fundamental matter hypermultiplet inserts a particular vertex operator in this theory. In this way we get a generalization of the main result of cite{CO} to $K$-theory. The theory of interpolating Macdonald polynomials is an important tool in our construction.
We find a limit formula for a generalization of MacDonalds inner product in finitely many variables, using equivariant localization on the Grassmannian variety, and the main lemma from cite{Car}, which bounds the torus characters of the higher c{C}ec
h cohomology groups. We show that the MacDonald inner product conjecture of type $A$ follows from a special case, and the Pieri rules section of MacDonalds book cite{Mac}, making this limit suitable replacement for the norm squared of one, the usual normalizing constant.
We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.
Given a finite subset S in F_p^d, let a(S) be the number of distinct r-tuples (x_1,...,x_r) in S such that x_1+...+x_r = 0. We consider the moments F(m,n) = sum_|S|=n a(S)^m. Specifically, we present an explicit formula for F(m,n) as a product of two
matrices, ultimately yielding a polynomial in q=p^d. The first matrix is independent of n while the second makes no mention of finite fields. However, the complexity of calculating each grows with m. The main tools here are the Schur-Weyl duality theorem, and some elementary properties of symmetric functions. This problem is closely to the study of maximal caps.
