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We provide several ingredients towards a generalization of the Littlewood-Richardson rule from Chow groups to algebraic cobordism. In particular, we prove a simple product-formula for multiplying classes of smooth Schubert varieties with…
A breakthrough in the theory of (type A) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for…
The Weyl-Kac character formula gives a beautiful closed-form expression for the characters of integrable highest-weight modules of Kac-Moody algebras. It is not, however, a formula that is combinatorial in nature, obscuring positivity. In…
We consider potential theory on Bratteli diagrams arising from Macdonald polynomials. The case of Hall-Littlewood polynomials is particularly interesting; the elements of the diagram are partitions, the branching multiplicities are…
We study two different one-parameter generalizations of Littlewood--Richardson coefficients, namely Hall polynomials and generalized inverse Kostka polynomials, and derive new combinatorial formulae for them. Our combinatorial expressions…
Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in…
In addition to extending some facts from field coefficients to commutative ring coefficients for Leavitt path algebras with new shorter proofs, we also prove some results that are new even for field coefficients. In particular, we show that…
In arXiv:1106.4305 extended superpolynomials were introduced for the torus links T[m,mk+r], which are functions on the entire space of time variables and, at expense of reducing the topological invariance, possess additional algebraic…
Using a finite-dimensional Clifford algebra a new combinatorial product formula for the small quantum cohomology ring of the complex Grassmannian is presented. In particular, Gromov-Witten invariants can be expressed through certain…
It is known that the usual Schur $S$- and $P$-polynomials can be described via the Gysin homomorphisms for flag bundles in the ordinary cohomology theory. Recently, P. Pragacz generalized these Gysin formulas to the Hall-Littlewood…
The ring of symmetric functions $\Lambda$, with natural basis given by the Schur functions, arise in many different areas of mathematics. For example, as the cohomology ring of the grassmanian, and as the representation ring of the…
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in \cite{Chi08}, we use quiver theory to represent the generalized Littlewood-Richardson coefficients for the branching rule for the diagonal embedding of $\gl(n)$ as…
We present a sum-product formula for the classical Hall polynomial which is based on tableaux that have been introduced by T. Klein in 1969. In the formula, each summand corresponds to a Klein tableau, while the product is taken over the…
Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the…
We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles, generalizing Gysin formulas for Schur S- and Q-functions.
We introduce elliptic weights of boxed plane partitions and prove that they give rise to a generalization of MacMahon's product formula for the number of plane partitions in a box. We then focus on the most general positive degenerations of…
We consider a subclass of tilings, the tilings obtained by cut and projection. Under somewhat standard assumptions, we show that the natural complexity function has polynomial growth. We compute its exponent \alpha in terms of the ranks of…
We give a concrete combinatorial interpretation of the coefficients of the Kazhdan-Lusztig polynomials of Dowling geometries, a family of matroids which generalizes braid matroids of types A and B. Furthermore, we interpret the coefficients…
In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by {\it Clifford cohomology.} We show that {\it Clifford…
Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.