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We consider an auxiliary spectral problem originally introduced by Gerdjikov, Mikhailov and Valchev (GMV system) and its modification called pseudo-Hermitian reduction which is extensively studied here for the first time. We describe the…

A set of functions is introduced which generalizes the famous Schur polynomials and their connection to Grasmannian manifolds. These functions are shown to provide a new method of constructing solutions to the KP hierarchy of nonlinear…

The Witten index counts the difference in the number of bosonic and fermionic states of a quantum mechanical system. The Schur index, which can be defined for theories with at least $\mathcal{N}=2$ supersymmetry in four dimensions is a…

We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. This expansion follows from several new…

It is well known that if one integrates a Schur function indexed by a partition $\lambda$ over the symplectic (resp. orthogonal) group, the integral vanishes unless all parts of $\lambda$ have even multiplicity (resp. all parts of $\lambda$…

We extend Schaumann's theory of pivotal structures on fusion categories matched to a module category and of module traces developed in arXiv:1206.5716 to the case of non-semisimple tensor categories, and use it to study eigenvalues of the…

We study the notion of weak amalgamation in the context of diagonal conjugacy classes. Generalizing results of Kechris and Rosendal, we prove that for every countable structure $M$, Polish group $G$ of permutations of $M$, and $n \geq 1$,…

We study structural properties of truncated Weyl modules. A truncated Weyl module $W_N(\lambda)$ is a local Weyl module for $\mathfrak g[t]_N = \mathfrak g \otimes \frac{\mathbb C[t]}{t^N\mathbb C[t]}$, where $\mathfrak g$ is a…

The dual immaculate functions are a basis of the ring QSym of quasisymmetric functions, and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating…

We construct the shifted genus expanded $\cal{W}_{\infty}$ algebra, which is isomorphic to the central subalgebra $\cal{A}_{\infty}$ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra $\Lambda^\ast$…

In this paper, we study the irreducibility of $\mathcal{U}(\mathfrak{g})^{G'}$-modules on the spaces of intertwining operators in the branching problem of reductive Lie algebras, and construct a family of finite-dimensional irreducible…

Recently, K. Bringmann, P. Guerzhoy, Z. Kent and K. Ono studied the connection between Eichler integrals and the holomorphic parts of harmonic weak Maass forms on the full modular group. In this article, we extend their result to more…

We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur…

We analyze the structure of the family of quadratic superalgebras, introduced in J Phys A 44(23):235205 (2011), for the quadatic deformations of $N = 1$ space-time conformal supersymmetry. We characterize in particular the `zero-step'…

We consider an invariant quantum Hamiltonian $H=-\Delta_{LB}+V$ in the $L^{2}$ space based on a Riemannian manifold $\tilde{M}$ with a countable discrete symmetry group $\Gamma$. Typically, $\tilde{M}$ is the universal covering space of a…

For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric function $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ a…

In this note we revive a transformation that was introduced by H. S. Wall and that establishes a one-to-one correspondence between continued fraction representations of Schur, Carath\'eodory, and Nevanlinna functions. This transformation…

Let $(G,G_1)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces $\mathfrak{p}^+_1\subset\mathfrak{p}^+$…

In 1952, J.H.Braun claimed to have established a formula giving a lower bound for certain partitions of sets of integers into weakly sum-free classes. However, no proof or supporting construction was published at that time. In today's…

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\text{GL}(n)$. The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition…