In the present paper we find a new interpretation of Narayana polynomials Nn(x) which are the generating polynomials for the Narayana numbers Nn,k = 1 n C k−1 n C k n where C i j stands for the usual binomial coefficient, i.e. C j = j! i!(j−i)! . They count Dyck paths of length n and with exactly k peaks, see e.g. [13] and they appeared recently in a number of different combinatorial situations...

Fix a rectangular Young diagram R, and consider all the products of Schur functions sλsλc , where λ and λ c run over all (unordered) pairs of partitions which are complementary with respect to R. Theorem: The self-complementary products, s2λ where λ = λ , are linearly independent of all other sλsλc . Conjecture: The products sλsλc are all linearly independent.

The higher order analogue of the classical Carathéodory-Julia theorem on boundary angular derivatives has been obtained in [7]. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in the open unit disk) motivated by that result. Mathematics Subject Classification (2000). 47A57, 47A20, 47A48.

Fix a rectangular Young diagram R, and consider all the products of Schur functions sλsλc , where λ and λ c run over all (unordered) pairs of partitions which are complementary with respect to R. Theorem: The self-complementary products, s λ where λ = λ, are linearly independent of all other sλsλc . Conjecture: The products sλsλc are all linearly independent.

For a particular set of Boltzmann weights and a particular boundary condition for the six vertex model in statistical mechanics, we compute explicitly the partition function and show it to be equal to a factorial Schur function, giving a new proof of a theorem of Lascoux.

We cover some of the foundational results of representation theory including Maschke’s Theorem, Schur’s Lemma, and the Schur Orthogonality Relations. We consider character theory, constructions of representations, and conjugacy classes. Finally, we touch upon Pontryagin Duality for abelian groups and Young Tableaux for the Symmetric Group.

Combinatorial formulas for the moments of the Brownian motion on classical compact Lie groups are obtained. These expressions are deformations of formulas of B. Collins and P. Śniady for moments of the Haar measure and yield a proof of the First Fundamental Theorem of Invariant and of classical Schur-Weyl dualities based on stochastic calculus.

We prove Okounkov’s conjecture, a conjecture of Fomin-FultonLi-Poon, and a special case of Lascoux-Leclerc-Thibon’s conjecture on Schur positivity and derive several more general statements using a recent result of Rhoades and Skandera. 1. Schur positivity conjectures The ring of symmetric functions has a linear basis of Schur functions sλ labelled by partitions λ = (λ1 ≥ λ2 ≥ · · · ≥ 0). A sym...

We give general integral formulas involving for hyperdeterminants or hyperpfaffians. In the applications, we obtain several summation formulas for the products of Schur functions, which are generalizations of Cauchy’s determinant. Further, we study Toeplitz hyperdeterminants by using the theory of Jack polynomials and give a hyperdeterminant version of a strong Szegö limit theorem. MSC-class: p...

Our main theorem, proved in § 2 establishes some of the properties of F(x, y) = 0 when we drop all restrictions on / and require only that all the roots of h be real (of arbitrary sign). In the last section, we apply this theorem to obtain an extension of Theorem 1.1 which states that, for h restricted as in Pόlya's theorem, there are at least as many intersection points as the number of real r...