Structure constants for the multiplication of Schubert polynomials by Schur symmetric polynomials are related to the enumeration of chains in a new partial order on S 1 , the Grassmaniann Bruhat order. Here we present a monoid M related to this order. We develop a notion of reduced sequences for M and show that M is analogous to the nil-Coxeter monoid for the weak order on S 1 .

We study conditions for a d –Kummer pullback of a Picard–Fuchs equation in a twisted symmetric square of the universal elliptic curve over X0(N) W to be of type D3. Our main result says that the only possible pairs (N, d) are exactly those for which there exists a Fano threefold with one-dimensional Picard group whose index is d and whose anticanonical degree is 2d2N. We give some modular formu...

We show that the Nakayama automorphism of a Frobenius algebra R over a field k is independent of the field (Theorem 4). Consequently, the k-dual functor on left R-modules and the bimodule isomorphism type of the k-dual of R, and hence the question of whether R is a symmetric k-algebra, are independent of k. We give a purely ring-theoretic condition that is necessary and sufficient for a finite-...

The Lie module of the group algebra FSn of the symmetric group is known to be not projective if and only if the characteristic p of F divides n. We show that in this case its nonprojective summands belong to the principal block of FSn. Let V be a vector space of dimension m over F , and let L(V ) be the n-th homogeneous part of the free Lie algebra on V ; this is a polynomial representation of ...

Abstract. Let X be a smooth scheme of finite type over a field K, let E be a locally free OX -bimodule of rank n, and let A be the non-commutative symmetric algebra generated by E. We construct an internal Hom functor, HomGrA(−,−), on the category of graded right A-modules. When E has rank 2, we prove that A is Gorenstein by computing the right derived functors of HomGrA(OX ,−). When X is a smo...

The eigenvalues of the elliptic N-body Ruijsenaars operator are obtained by a dynamical version of the algebraic nested Bethe ansatz method. The result is derived by using the construction given in [1], where the Ruijsenaars operator was obtained as the transfer matrix associated to the symmetric power of the vector representation of the elliptic quantum group Eτ,γ(glN ).

The nucleus of a Veronese variety is the intersection of all its osculating hyperplanes. Various authors have given necessary and sufficient conditions for the nucleus to be empty. We present an explicit formula for the dimension of this nucleus for arbitrary characteristic of the ground field. As a corollary, we obtain a dimension formula for that subspace in the t–th symmetric power of a fini...

In this paper, we consider a family of symmetric polynomials of the eigenvalues of a complex matrix A and find an explicit expression of each member of the family as a polynomial of the entries of A with positive coefficients. In the case of a nonnegative matrix, one immediately obtains a family of inequalities involving matrix eigenvalues and diagonal entries. Equivalent forms of some of the o...