Related papers: A combinatorial $\mathfrak{sl}_2$-action and the S…

Motivated by Stanley's conjecture on the multiplication of Jack symmetric functions, we prove a couple of identities showing that skew Jack symmetric functions are semi-invariant up to translation and rotation of a $\pi$ angle of the skew…

We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n,C). Our main results are: 1) Pieri rules for the Schubert bases of H^*(Gr) and H_*(Gr), which expresses the product of a special…

Recently, Chu studied some properties of the partial sums of the sequence $P^k(F_n)$, where $P(F_n)=\big(\sum_{i=1}^nF_i\big)_{n\geq1}$ and $(F_n)_{n\geq1}$ is the Fibonacci sequence, and gave its combinatorial interpretation. We generalize…

Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight \lambda, we define a preorder on the set $P(\lambda, k)$ of k-tuples of dominant weights which add up to \lambda. Let $P(\lambda,…

We find an explicit combinatorial gradient vector field on the well known complex S (Salvetti complex) which models the complement to an arrangement of complexified hyperplanes. The argument uses a total ordering on the facets of the…

This paper generalizes the basic notions of additive and multiplicative combinatorics to the setting of group actions: if $G$ is a group acting on a set $X$, and we have subsets $A\subseteq G$ and $Y\subseteq X$ such that the set of pairs…

We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call \emph{weak Feller property}. Our characterization involves potential theoretic as well as probabilistic aspects…

A recent result of Balandraud shows that for every subset S of an abelian group G, there exists a non trivial subgroup H such that |TS| <= |T|+|S|-2 holds only if the stabilizer of TS contains H. Notice that Kneser's Theorem says only that…

It is shown that a contraction on a Hilbert space is complex symmetric if and only if the values of its characteristic function are all symmetric with respect to a fixed conjugation. Applications are given to the description of complex…

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are noetherian whenever R is ; a…

Stanley's inequalities for partially ordered sets establish important log-concavity relations for sequences of linear extensions counts. Their extremals however, i.e., the equality cases of these inequalities, were until now poorly…

We construct Luzin-type subsets of the real line in all finite powers Rothberger, with a non-Menger product. To this end, we use a purely combinatorial approach which allows to weaken assumptions used earlier to construct sets with…

We show that the Coxeter-sortable elements in a finite Coxeter group W are the minimal congruence-class representatives of a lattice congruence of the weak order on W. We identify this congruence as the Cambrian congruence on W, so that the…

For an $n$-tuple $s$ of non-negative integers, the $s$-weak order is a lattice structure on $s$-trees, generalizing the weak order on permutations. We first describe the join irreducible elements, the canonical join representations, and the…

With the help of a useful mathematical tool, the polar decomposition of closed operators, and a simple observation, i.e. the unique relation between tensor-product states and compact operators, we manage to give a compact and coherent…

We prove the existence of signed combinatorial interpretations for several large families of structure constants. These families include standard bases of symmetric and quasisymmetric polynomials, as well as various bases in Schubert…

The study of pinnacle sets has been a recent area of interest in combinatorics. Given a permutation, its pinnacle set is the set of all values larger than the values on either side of it. Largely inspired by conjectures posed by Davis,…

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the ``order formula'' of D. Testerman…

Tukey order are used to compare the cofinal complexity of partially order sets (posets). We prove that there is a $2^\mathfrak{c}$-sized collection of sub-posets in $2^\omega$ which forms an antichain in the sense of Tukey ordering. Using…

This paper is, essentially, a survey related to the problem of understanding the combinatorics of the action of the monoidal category of finite dimensional modules over a simple finite dimensional Lie algebra on various categories of Lie…