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We present families of single determinantal representations of on-shell scalar products of Bethe vectors. Our families of representations are parameterized by a continuous complex variable which can be fixed at convenience. Here we consider…

An h-tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the…

We study $h$-vectors of simplicial complexes which satisfy Serre's condition ($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for all…

We describe some of the basic properties of the 2-category of 2-term complexes in an abelian category, using butterflies as morphisms.

In this article, we study the Schur mutiplier of the discrete as well as the finite Heisenberg groups and their t-variants. We describe the representation groups of these Heisenberg groups and through these give a construction of their…

We consider unimodality and related properties of f-vectors of polytopes in various dimensions. By a result of Kalai (1988), f-vectors of 5-polytopes are unimodal. In higher dimensions much less can be said; we give an overview on current…

We study the structure of two-sided vector spaces over a perfect field $K$. In particular, we give a complete characterization of isomorphism classes of simple two-sided vector spaces which are left finite-dimensional. Using this…

We determine the set of the Bloch vectors for N-level systems, generalizing the familiar Bloch ball in 2-level systems. An origin of the structural difference from the Bloch ball in 2-level systems is clarified.

We prove an explicit formula for a projection of singular vectors in the Verma module over a rank 2 Kac-Moody Lie algebra onto the universal enveloping algebra of the Heisenberg Lie algebra and of $sl_{2}$ (Theorem 3). The formula is…

In this paper, we answer two questions on local $h$-vectors, which were asked by Athanasiadis. First, we characterize all possible local $h$-vectors of quasi-geometric subdivisions of a simplex. Second, we prove that the local…

Presented is a free boson representation of the type II vertex operators for the $A_{n-1}^{(1)}$ face model. Using the bosonization, we derive some properties of the type II vertex operators, such as commutation, inversion and duality…

We characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the second author.

Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if…

We present the characteristic polynomial for the transition matrix of a vertex-face walk on a graph, and obtain its spectra. Furthermore, we express the characteristic polynomial for the transition matrix of a vertex-face walk on the…

Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric…

Let ${\mathfrak{g}}$ be a complex semisimple Lie algebra with Borel subalgebra ${\mathfrak{b}}$ and corresponding nilradical ${\mathfrak{n}}$. We show that singular Whittaker modules $M$ are simple if and only if the space $\hbox{Wh}\,M$ of…

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or,…

In this article we describe the 2-cocycles, Schur multiplier and representation group of discrete Heisenberg groups over the unital rings of order $p^2$. We describe all projective representations of Heisenberg groups with entries from the…

We classify rank two vector bundles on P3 with Buchsbaum index equal to three and also give some results on the H1-module of "negative instanton"bundles.

We introduce braid monodromy for the discriminant hypersurface in versal unfoldings of hypersurface singularities. Our objective is then to compute this invariant for singularities of Brieskorn Pham type: First we consider the unfolding by…