Related papers: Symmetric products, linear representations and the…

It is well-known that the homology of the classifying space of the unitary group is isomorphic to the ring of symmetric functions, Symm. We offer the cohomology of the loop space of the suspension of the infinite complex projective space as…

In this paper we study the ring $\mathcal{P}$ of combinatorial convex polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all its…

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…

We show that integration over a $G$-manifold $M$ can be reduced to integration over a minimal section $\Sigma$ with respect to an induced weighted measure and integration over a homogeneous space $G/N$. We relate our formula to integration…

The_commuting variety_ is the pairs of NxN matrices (X,Y) such that XY = YX. We introduce the_diagonal commutator scheme_, {(X,Y) : XY-YX is diagonal}, which we prove to be a reduced complete intersection, one component of which is the…

We compute all massive partition functions or characteristic polynomials and their complex eigenvalue correlation functions for non-Hermitean extensions of the symplectic and chiral symplectic ensemble of random matrices. Our results are…

We study two new families of symmetric functions arising from a species-theoretic construction motivated by cycle structure. For each partition of $n$, we define two combinatorial species that decompose into molecules indexed by the same…

We study the problem of describing the set of real functionals on the quotient $\textrm{Sym}/(p_2-1)$ of the ring of symmetric functions that are nonnegative on the images of certain modified Hall-Littlewood symmetric functions. This…

For a polynomial ring over a commutative ring of positive characteristic, we define on the associated de Rham-Witt complex a set of functions, and show that they are pseudovaluations in the sense of Davis, Langer and Zink. To achieve it, we…

Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect…

The concept of rotation symmetric functions from the Boolean domain is extended to the multiple-valued (MV) domain. It is shown that symmetric functions are a subset of the rotation symmetric functions. Functions exhibiting these kinds of…

Let $V=V_1 \otimes \cdots \otimes V_n$ be a vector space over an algebraically closed field $K$ of characteristic zero with $\dim(V_i)=d_i$. We study the ring of polynomial invariants $K[\operatorname{End}(V)^{\oplus…

In this paper, using the theory of category, we generalize known properties of symmetric polynomials and functions and characterize the multi-indicial symmetric functions. Examples have been given on Schur functions.

We study the ring of quasisymmetric polynomials in $n$ anticommuting (fermionic) variables. Let $R_n$ denote the polynomials in $n$ anticommuting variables. The main results of this paper show the following interesting facts about…

The aim of this note is to introduce a compound basis for the space of symmetric functions. Our basis consists of products of Schur functions and $Q$-functions. The basis elements are indexed by the partitions. It is well known that the…

In this paper, we study harmonic analysis on finite homogeneous spaces whose associated permutation representation decomposes with multiplicity. After a careful look at Frobenius reciprocity and transitivity of induction, and the…

A function $f$ from $\mathbb{Z}$ to the symmetric matrices over an arbitrary field $K$ of characteristic $0$ is a $1$-quasihomomorphism if the matrix $f(x+y) - f(x) - f(y)$ has rank at most $1$ for all $x,y \in \mathbb{Z}$. We show that any…

The connection between the commutativity of a family of $n\times n$ matrices and the generalized joint numerical ranges is studied. For instance, it is shown that ${\cal F}$ is a family of mutually commuting normal matrices if and only if…

We construct a symmetric invertible binary pairing function $F(m,n)$ on the set of positive integers with a property of $F(m,n)=F(n,m)$. Then we provide a complete proof of its symmetry and bijectivity, from which the construction of…

Given a ring $R$, we study the bimodules $M$ for which the trivial extension $R\propto M$ is morphic. We obtain a complete characterization in the case where $R$ is left perfect, and we prove that $R\propto Q/R$ is morphic when $R$ is a…