Introduced by Okounkov and Reshetikhin in 2003, the Schur Process has been shown to be a determinantal point process, so that each of its correlation functions are determinants of minors of one correlation kernel matrix. In previous papers, this was derived using determinantal expressions of the skew-Schur functions; in this paper, we obtain this result in a different way, using the fact that t...

We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C∗-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we es...

This paper is focused on the applications of Schur complements to determinant inequalities. It presents a monotonic characterization of Schur complements in the L öwner partial ordering sense such that a new proof of the Hadamard-Fischer-Koteljanski inequality is obtained. Meanwhile, it presents matrix identities and determinant inequalities involving positive semidefinite matrices and extends ...

The paper compares coupled multigrid methods and pressure Schur complement schemes (operator splitting schemes) for the solution of the steady state and time dependent incompressible Navier–Stokes equations. We consider pressure Schur complement schemes with multigrid as well as single grid methods for the solution of the Schur complement problem for the pressure. The numerical tests have been ...

We give a presentation of cyclotomic q-Schur algebras by generators and defining relations. As an application, we give an algorithm for computing decomposition numbers of cyclotomic q-Schur algebras. § 0. Introduction Let Hn,r be an Ariki-Koike algebra associated to a complex reflection group Sn ⋉ (Z/rZ). A cyclotomic q-Schur algebra Sn,r associated to Hn,r, introduced in [DJM], is defined as a...

‎the global generalized minimum residual (gl-gmres)‎ ‎method is examined for solving the generalized sylvester matrix equation‎ ‎[sumlimits_{i = 1}^q {a_i } xb_i = c.]‎ ‎some new theoretical results are elaborated for‎ ‎the proposed method by employing the schur complement‎. ‎these results can be exploited to establish new convergence properties‎ ‎of the gl-gmres method for solving genera...

We provide direct proofs of product and coproduct formulae for Schur functions where the coefficients (Littlewood–Richardson coefficients) are defined as counting puzzles. The product formula includes a second alphabet for the Schur functions, allowing in particular to recover formulae of [Molev–Sagan ’99] and [Knutson–Tao ’03] for factorial Schur functions. The method is based on the quantum i...

In recent years, there has been considerable interest in showing that certain conditions on skew shapes A and B are sufficient for the difference sA − sB of their skew Schur functions to be Schur-positive. We determine necessary conditions for the difference to be Schur-positive. Our conditions are motivated by those of Reiner, Shaw and van Willigenburg that are necessary for sA = sB , and we d...

In this paper, we show the relation between the Schur algebras Sr Λ,Σ(B) and S r′ Λ,Σ(B), where 1 ≤ r ′ < r < ∞. Then we set up the involution operator in these Schur algebras and show that with this involution operator there is only one C∗-algebra among these classes of Banach algebras. Furthermore, we show the equivalence of a condition on the Schur multiplier norm and the existence of C∗-alg...