The universal character is a generalization of the Schur polynomial attached to a pair of partitions; see [8]. We prove that the universal character solves the Darboux chain. The N -periodic closing of the chain is equivalent to the Painlevé equation of type A N−1. Consequently we obtain an expression of rational solutions of the Painlevé equations in terms of the universal characters. Introduc...

Abstract We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) functors give new interpretation representations the general linear groups GL n , (polynomial) symmetric pair ( $$ {U}_{Q,q}^B U Q , q ...

It is well known that the C∗-algebra of an ordered pair of qubits is M2⊗M2. What about unordered pairs? We show in detail that M3⊕C is the C∗-algebra of an unordered pair of qubits. Then we use Schur-Weyl duality to characterize the C∗-algebra of an unordered n-tuple of d-level quantum systems. Using some further elementary representation theory and number theory, we characterize the quantum cy...

We define a unified trace form τ on the cyclotomic Hecke algebras Hn,K of type A, which generalize both Malle-Mathas' non-degenerate version (with parameter ξ≠1) and Brundan-Kleshchev's degenerate version. use seminormal basis theory to construct an explicit pair dual bases for with respect form. also cocenter corresponding Schur algebra, shows that has dimension independent ground field K, ξ p...

We discuss Arens regularity of the projective tensor product Banach algebras consisting Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space ℋ, Sp(ℋ)⊗γ Sq (ℋ) and B(S2(ℋ))⊗γ S2(ℋ) are not regular every pair 1≤p,q≤2; whereas, when ℋ is separable equipped with Schur product, show completely continuous S2(ℋ)⊗γ regular.

The shifted Schur measure introduced in [TW2] is a measure on the set of all strict partitions λ = (λ1 > λ2 > · · · > λl > 0), which is defined by Schur Q-functions. The main aim of this paper is to calculate the correlation function of this measure, which is given by a pfaffian. As an application, we prove that a limit distribution of λj ’s with respect to a shifted version of the Plancherel m...

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (− ‖−) on any pair of “colored” almost positive real Schur roots which generalizes previous de...

We prove that a certain sequence of tau functions of the Garnier system satisfies Toda equation. We construct a class of algebraic solutions of the system by the use of Toda equation; then show that the associated tau functions are expressed in terms of the universal character, which is a generalization of Schur polynomial attached to a pair of partitions. This article is based on the results i...

Let F be an arbitrary field (of arbitrary characteristic). Let A be a central simple F -algebra endowed with a quadratic pair σ (if charF 6= 2 then σ is simply an orthogonal involution on A). We show that the Witt index of σ over the function field of the Severi-Brauer variety of A is divisible by the Schur index of the algebra A.

We study the q-difference analogue of the sixth Painlevé equation (q-PVI) by means of tau functions associated with affine Weyl group of type D5. We prove that a solution of q-PVI coincides with a self-similar solution of the q-UC hierarchy. As a consequence, we obtain in particular algebraic solutions of q-PVI in terms of the universal character which is a generalization of Schur polynomial at...