The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functio...

We introduce the notion of a Schur-finite element in a λ-ring. Since the beginning of algebraic K-theory in [G57], the splitting principle has proven invaluable for working with λ-operations. Unfortunately, this principle does not seem to hold in some recent applications, such as the K-theory of motives. The main goal of this paper is to introduce the subring of Schur-finite elements of any λ-r...

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...

The celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give an algebraic and a combinatorial proof of Naruse’s formula, by using factorial Schur functio...

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...

DIRECTION-PRESERVING AND SCHUR-MONOTONIC SEMISEPARABLE APPROXIMATIONS OF SYMMETRIC POSITIVE DEFINITE MATRICES∗ MING GU† , XIAOYE S. LI‡ , AND PANAYOT S. VASSILEVSKI§ Abstract. For a given symmetric positive definite matrix A ∈ RN×N , we develop a fast and backward stable algorithm to approximate A by a symmetric positive definite semiseparable matrix, accurate to a constant multiple of any pres...

1 , . . . , xn, x −1 n factorizes into a product of two odd orthogonal characters of rectangular shape, one of which is evaluated at −x1, . . . ,−xn, if M is even, while it factorizes into a product of a symplectic character and an even orthogonal character, both of rectangular shape, if M is odd. It is furthermore shown that the first factorization implies a factorization theorem for rhombus t...

In a recent paper, [2], Gill, using methods of the modular representation theory of finite groups, describes some results on the tensor product of Young modules for symmetric groups. We here give an alternative approach using the polynomial representation theory of general linear groups and the Schur functor. The main result is a formula for the multiplicities of Young modules in a tensor produ...

In the present paper, we first prove the logarithmic convexity of the elementary function b x −a x x , where x 6= 0 and b > a > 0. Basing on this, we then provide a simple proof for Schur-convex properties of the extended mean values, and, finally, discover some convexity related to the extended mean values.

We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and ∏ 1≤i<j≤n(xi + yj). This result generalises a number of well–known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)standard tableaux whi...