The isomorphism problem for finitely generated Coxeter groups is the problem of deciding if two finite Coxeter matrices define isomorphic Coxeter groups. Coxeter [3] solved this problem for finite irreducible Coxeter groups. Recently there has been considerable interest and activity on the isomorphism problem for arbitrary finitely generated Coxeter groups. In this paper, we determine some stro...

The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left-right eigenvectors of a matrix C. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of p-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left-right p-dimen...

Introduction. Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of G of JX up to isogeny. The most prominent example of such a situation is the case of the group G ≃ Z/2Z of two elements. Let π : X → Y = X/G denote the canonical quotient map. The first to notice that JX is isogenous to the product JY × P (...

The two-sided Rayleigh quotient iteration proposed by Ostrowski computes a pair of corresponding left-right eigenvectors of a matrix C. We propose a Grassmannian version of this iteration, i.e., its iterates are pairs of p-dimensional subspaces instead of one-dimensional subspaces in the classical case. The new iteration generically converges locally cubically to the pairs of left-right p-dimen...

In this paper, we describe the irreducible spherical functions of fundamental K-types associated with the pair (G,K) = (SO(n+ 1),SO(n)) in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same K-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size 2 and 3, w...

Example. Let X be an algebraic projective irreducible surface over a field k and let P be a closed point of X , C ⊂ X be an irreducible curve such that P ∈ C . If X and C are smooth at P , then we let t ∈ OX,P be a local equation of C at P and u ∈ OX,P be such that u|C ∈ OC,P is a local parameter at P . Denote by C the ideal defining the curve C near P . Now we can introduce a two-dimensional l...

The symbol-pair codes over finite fields have been raised for symbol-pair read channels and motivated by application of high-density data storage technologies [1, 2]. Their generalization is the code for b-symbol read channels (b > 2). Many MDS codes for b-symbol read channels have been constructed which meet the Singleton-like bound ([3, 4, 10] for b = 2 and [11] for b > 2). In this paper we s...

The Grothendieck ring of k-varieties is still very poorly understood. In characteristic zero, the quotient of K0[Vark] by the ideal generated by [A ] is naturally isomorphic to the ring Z[SBk], where Z[SBk] is the free abelian group generated by the stable birational equivalence classes of smooth, projective, irreducible k-varieties and multiplication is given by the product of varieties [Lar-L...

Let π1,…,πk be smooth irreducible representations of p-adic general linear groups. We prove that the parabolic induction product π1×⋯×πk has a unique quotient whose Langlands parameter is sum parameters all factors (cyclicity property), assuming same property holds for each products πi×πj (i<j), and but at most two πi×πi remains (square-irreducibility property). Our technique applies recently d...

It was conjectured by Haiman [H] that the space of diagonal coinvariants for a root system R of rank n has a ”natural” quotient of dimension (1 + h) for the Coxeter number h. This space is the quotient C[x, y]/(C[x, y]C[x, y]o ) for the algebra of polynomials C[x, y] with the diagonal action of the Weyl group on x ∈ C ∋ y and the ideal C[x, y]o ⊂ C[x, y] of the W -invariant polynomials without ...