We use entropy rates and Schur concavity to prove that, for every integer k ≥ 2, every nonzero rational number q, and every real number α, the base-k expansions of α, q + α, and qα all have the same finite-state dimension and the same finitestate strong dimension. This extends, and gives a new proof of, Wall’s 1949 theorem stating that the sum or product of a nonzero rational number and a Borel...

We obtain a presentation by generators and relations for generalized Schur algebras and their quantizations. This extends earlier results obtained in the type A case. The presentation is compatible with Lusztig’s modified form : U of a quantized enveloping algebra. We show that generalized Schur algebras inherit a canonical basis from : U, that this gives them a cellular structure, and thus the...

The eigenvalues of a self-adjoint n×n matrix A can be put into a decreasing sequence λ = (λ1, . . . , λn), with repetitions according to multiplicity, and the diagonal of A is a point of R that bears some relation to λ. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We prove an extension of the latter result for positive trace-class operators on ...

The eigenvalues of a self-adjoint n×n matrix A can be put into a decreasing sequence λ = (λ1, . . . , λn), with repetitions according to multiplicity, and the diagonal of A is a point of R that bears some relation to λ. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We prove an extension of the latter result for positive trace-class operators on ...

We consider compactifications of (Pn)3\ ⋃ ∆ij , the space of triples of distinct points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple. We compute the sections of line bundles on these spaces, and show that they are equal as ...

The eigenvalues of a self-adjoint n×n matrix A can be put into a decreasing sequence λ = (λ1, . . . , λn), with repetitions according to multiplicity, and the diagonal of A is a point of R that bears some relation to λ. The Schur-Horn theorem characterizes that relation in terms of a system of linear inequalities. We give a new proof of the latter result for positive trace-class operators on in...

Let n ≥ 4 be an even integer. Let K be a field with charK 6= 2 and q an invertible element in K such that Qn−1 i=1 (1+q ) 6= 0. In this paper, we study the decomposition numbers over K of the Iwahori–Hecke algebra Hq(Dn) of type Dn. We obtain some equalities which relate its decomposition numbers with certain Schur elements and the decomposition numbers of various Iwahori–Hecke algebras of type...

We extend the Carathéodory–Julia theorem on angular derivatives as well as its higher order analogue established recently in [4] to the setting of contractive valued functions analytic on the unit disk. Carathéodory–Julia type conditions for an operator valued Schur-class function w are shown to be equivalent to the requirement that every function from the de Branges-Rovnyak space associated wi...

The Schur-Weyl Duality Theorem motivates the general deenition of dual representations. Such representations arise in a very large class of groups, including all compact, nilpotent and semidirect product groups, as well as the complementary groups in the physics literature and the reductive dual pairs in the mathematics literature. Several applications of the notion of duality are given, includ...

In this paper we describe planar decompositions of skew shape tableaux into strips and use the shapes of these strips to generate a determinant. We then prove that each of these determinants is equal to the Schur function for the skew shape. The Jacobi-Trudi identity, the dual Jacobi-Trudi identity, the Giambelli identity and the rim ribbon identity of Laseoux and Pragacz are all special cases ...