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...

For x x1, x2, . . . , xn ∈ 0, 1 n and r ∈ {1, 2, . . . , n}, the symmetric function Fn x, r is defined as Fn x, r Fn x1, x2, . . . , xn; r ∑ 1≤i1<i2 ···<ir≤n ∏r j 1 1 xij / 1−xij , where i1, i2, . . . , in are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of Fn x, r are discussed. As consequences, several inequalities are est...

A theorem due to Tokuyama expresses Schur polynomials in terms of GelfandTsetlin patterns, providing a deformation of the Weyl character formula and two other classical results, Stanley’s formula for the Schur q-polynomials and Gelfand’s parametrization for the Schur polynomials. We generalize Tokuyama’s formula to the Hall-Littlewood polynomials by extending Tokuyama’s statistics. Our result, ...

For x x1, x2, . . . , xn ∈ R , the symmetric function φn x, r is defined by φn x, r φn x1, x2, . . . , xn; r ∏ 1≤i1<i2 ···<ir≤n ∑r j 1 xij / 1 xij 1/r , where r 1, 2, . . . , n and i1, i2, . . . , in are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of φn x, r are discussed. As applications, some inequalities are established...

Multi-Schur functions are symmetric that generalize the supersymmetric Schur functions, flagged and refined dual Grothendieck which have been intensively studied by Lascoux. In this paper, we give a new free-fermionic presentation of them. The multi-Schur indexed partition two ``tuples tuples'' indeterminates. We construct family linear bases fermionic Fock space such data prove they correspond...

The formula (1) was first used by Schur [22], but the idea of the Schur complement goes back to Sylvester (1851), and the term Schur complement was introduced by E. Haynsworth [16]. In the beginning Schur complements were used in the theory of matrices. M.G. Krein [19] and W.N. Anderson and G.E. Trapp [4] extended the notion of Schur complements of matrices to shorted operators in Hilbert space...

This paper uses the theory of dual equivalence graphs to give explicit Schur expansions to several families of symmetric functions. We begin by giving a combinatorial definition of the modified Macdonald polynomials and modified Hall-Littlewood polynomials indexed by any diagram δ ⊂ Z × Z, written as H̃δ(X; q, t) and P̃δ(X; t), respectively. We then give an explicit Schur expansion of P̃δ(X; t) as...

The Schur--Parlett algorithm, implemented in MATLAB as \textttfunm, evaluates an analytic function $f$ at $n\times n$ matrix argument by using the Schur decomposition and a block recurrence of P...