相关论文: The cd-index of the Boolean lattice

We prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean (real or complex) vector lattice, from which the Cauchy-Schwarz inequality follows. A reformulation of this result…

In this paper, we study flag structures of matroid base polytopes. We describe faces of matroid base polytopes in terms of matroid data, and give conditions for hyperplane splits of matroid base polytopes. Also, we show how the cd-index of…

Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and…

We introduce new classes of integrable models that exhibit a structure similar to that of flag vector spaces. We present their Hamiltonians, R-matrices and Bethe-ansatz solutions. These models have a new type of generalized graded algebra…

Let $\Lb$ be a lattice in an $n$-dimensional Euclidean space $E$ and let $\Lb'$ be a Minkowskian sublattice of $\Lb$, that is, a sublattice having a basis made of representatives for the Minkowski successive minima of $\Lb$. We consider the…

We compute the $cd$-index $\Psi_{cd}$ of matroid base polytopes $\mathscr{P}(M)$ for a large family of matroids $M$. The $cd$-index is a polynomial in two non-commutative variables that compactly encodes the count of face flags $\mathcal{F}…

We consider $d$-dimensional lattice polytopes $\Delta$ with $h^*$-polynomial $h^*_\Delta=1+h_k^*t^k$ for $1<k<(d+1)/2$ and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $1+h_k^*=p^r$ where $p$ is a prime number. These…

New identities on traces of representations of the Hecke algebra on the spaces of paths on graphs are presented. These identities are relevant in the computation of partition functions with fixed boundary conditions and of two-point…

A filter lattice is a distributive lattice formed by all filters of a poset in the anti-inclusion order. We study the combinatorial properties of the Hasse diagrams of filter lattices of certain posets, so called Fibonacci-like cubes, in…

Let M be a matroid on E, representable over a field of characteristic zero. We show that h-vectors of the following simplicial complexes are log-concave: 1. The matroid complex of independent subsets of E. 2. The broken circuit complex of…

This paper introduces a new and simple statistic on noncrossing partitions that expresses each coordinate of the toric $h$-vector of a cubical complex, written in the basis of the Adin $h$-vector entries, as the total weight of all…

We prove that the $\gamma$-vector of the barycentric subdivision of a simplicial sphere is the $f$-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used…

Using Ohkawa's theorem that the collection of Bousfield classes is a set, we perform a number of constructions with Bousfield classes. In particular, we describe a greatest lower bound operator; we also note that a certain subset DL of the…

We study a one-parameter family of vector-valued polynomials associated to each simple Lie algebra. When this parameter $q$ equals -1 one recovers Joseph polynomials, whereas at $q$ cubic root of unity one obtains ground state eigenvectors…

We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine…

In [4] we introduce the associative algebras $Q_{n,k}(\CE,\tau)$. Recall the definition. These algebras are labeled by discrete parameters $n,k$; $n,k$ are integers $n>k>0$ and $n$ and $k$ have not common divisors. Then, $\CE$ is an…

Motivated by the theory of Painlev\'e equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of…

We extend the definition of coarse flag Hilbert--Poincar\'e series to matroids; these series arise in the context of local Igusa zeta functions associated to hyperplane arrangements. We study these series in the case of oriented matroids by…

We continue the investigation of tabular algebras with trace (a certain class of associative ${\Bbb Z}[v, v^{-1}]$-algebras equipped with distinguished bases) by determining the extent to which the tabular structure may be recovered from a…

Interior and exterior angle vectors of polytopes capture curvature information at faces of all dimensions and can be seen as metric variants of $f$-vectors. In this context, Gram's relation takes the place of the Euler--Poincar\'e relation…