Vanishing theorems for tensor powers of an ample vector bundle

Summary LetE be a holomorphic vector bundle of rankr on a compact complex manifoldX of dimensionn. It is shown that the cohomology groupsH ^p,q (X, E^k (detE) ^l ) vanish ifE is ample andp+qn+1,ln–p+r–1. The proof rests on the well-known fact that every tensor powerE ^k , splits into irreducible representations of Gl(E). By Bott's theory, each component is canonically isomorphic to the direct image onX of a homogeneous line bundle over a flag manifold ofE. The proof is then reduced to the Kodaira-Akizuki-Nakano vanishing theorem for line bundles by means of the Leray spectral sequence, using backward induction onp. We also obtain a generalization of Le Potier's isomorphism theorem and a counterexample to a vanishing conjecture of Sommese.     

Hopf subalgebras and tensor powers of generalized permutation modules

By means of a certain module V$V$ and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R$R$ of a finite-dimensional Hopf algebra H$H$ is finite. The module V$V$ is the counit representation induced from R$R$ to H$H$, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V$V$ is either semisimple with R^∗$R^{\ast }$ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R$R$, then the depth of R$R$ in H$H$ is finite. One assigns a nonnegative integer depth to V$V$, or any other H$H$-module, by comparing the truncated tensor algebras of V$V$ in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character.     

Multiplicity function for tensor powers of modules of the A n algebra

We consider the decomposition of the pth tensor power of the module $L^{\omega _1 }$ over the algebra A_n into irreducible modules, $(L^{\omega _1 } )^{ \otimes p} = \sum\nolimits_v {m(v,p)L^v }$ . This problem occurs, for example, in finding the spectrum of an invariant Hamiltonian of a spin chain with p nodes. To solve the problem, we propose using the Weyl symmetry properties. For constructing the coefficients m(??, p) as functions of p, we develop an algorithm applicable to powers of an arbitrary module. We explicitly write an expression for the multiplicities m(??, p) in the decomposition of powers of the first fundamental module of sl(n+1). Based on the obtained results, we find new properties of systems of orthogonal polynomials (multivariate Chebyshev polynomials). Our algorithm can also be applied to tensor powers of modules of other simple Lie algebras.     

Riemann-Roch for tensor powers

Mapping a locally free module on a scheme to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behavior of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups. Received February 13, 1998; in final form January 12, 1999     

Independent sets in tensor graph powers

The tensor product of two graphs, G and H, has a vertex set V(G) × V(H) and an edge between (u,v) and (u′,v′) iff both u u′ ∈ E(G) and v v′ ∈ E(H). Let A(G) denote the limit of the independence ratios of tensor powers of G, lim, α(Gⁿ)/|V(Gⁿ)|. This parameter was introduced in [Brown, Nowakowski, Rall, SIAM J Discrete Math 9 ( 5 ), 290–300], where it was shown that A(G) is lower bounded by the vertex expansion ratio of independent sets of G. In this article we study the relation between these parameters further, and ask whether they are in fact equal. We present several families of graphs where equality holds, and discuss the effect the above question has on various open problems related to tensor graph products. © 2006 Wiley Periodicals, Inc. J Graph Theory     

The Schur functor on tensor powers

Let M be a left module for the Schur algebra S(n, r), and let \({s \in \mathbb{Z}^+}\) . Then \({M^{\otimes s}}\) is a \({(S(n,\,rs), F{\mathfrak{S}_{s}})}\) -bimodule, where the symmetric group \({{\mathfrak{S}_s}}\) on s letters acts on the right by place permutations. We show that the Schur functor f _rs sends \({M^{\otimes s}}\) to the \({(F{\mathfrak{S}_{rs}},F{\mathfrak{S}_s})}\) -bimodule \({F\mathfrak{S}_{rs}\otimes_{F(\mathfrak{S}_{r}\wr{\mathfrak{S}_s})} ((f_rM)^{\otimes s}\otimes_{F} F{\mathfrak{S}_s})}\) . As a corollary, we obtain the image under the Schur functor of the Lie power L ^s (M), exterior power \({\bigwedge^s(M)}\) of M and symmetric power S ^s (M).     

Variation of symmetric tensor powers and permanents

The norm of the derivative of the symmetric tensor power of an operator is evaluated exactly. A bound for the distance between the permanents of two matrices is obtained as a consequence.     

Projective modules and tensor products

In this paper we analyze the problem of transforming one path in Rⁿ to another by means of three geometric operations. The problem is approached in two different ways: via the theory of δ-indecomposable semigroups, and by means of combinatorics.     

Syzygies and tensor product of modules

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$ -modules. Assume $n$ is a nonnegative integer and that the tensor product $M\otimes _{R}N$ is an $(n+c)$ th syzygy of some finitely generated $R$ -module. If ${{\mathrm{Tor}}}^{R}_{>0}(M,N)=0$ , then both $M$ and $N$ are $n$ th syzygies of some finitely generated $R$ -modules.     

Vanishing of the top local cohomology modules over Noetherian rings

Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let be an ideal of R and denote the intersection of all prime ideals . It is shown that

Derivatives for antisymmetric tensor powers and perturbation bounds

In an earlier paper (R. Bhatia, T. Jain, Higher order derivatives and perturbation bounds for determinants, Linear Algebra Appl. 431 (2009) 2102-2108) we gave formulas for derivatives of all orders for the map that takes a matrix to its determinant. In this paper we continue that work, and find expressions for the derivatives of all orders for the antisymmetric tensor powers and for the coefficients of the characteristic polynomial. We then evaluate norms of these derivatives, and use them to obtain perturbation bounds.     

Lower bound inequalities for norms of symmetrized tensor powers

Let V be a complex inner product space of positive dimension m with inner product 〈·,·〉, and let T_n(V) denote the set of all n-linear complex-valued functions defined on V×V×?×V (n-copies). By S_n(V) we mean the set of all symmetric members of T_n(V). We extend the inner product, 〈·,·〉, on V to T_n(V) in the usual way, and we define multiple tensor products A₁⊗A₂⊗?⊗A_n and symmetric products A₁·A₂?A_n, where q₁,q₂,…,q_n are positive integers and A_i∈T_qi(V) for each i, as expected. If A∈S_n(V), then A^k denotes the symmetric product A·A?A where there are k copies of A. We are concerned with producing the best lower bounds for ‖A^k‖², particularly when n=2. In this case we are able to show that ‖A^k‖² is a symmetric polynomial in the eigenvalues of a positive semi-definite Hermitian matrix, M_A, that is closely related to A. From this we are able to obtain many lower bounds for ‖A^k‖². In particular, we are able to show that if ω denotes 1/r where r is the rank of M_A, and , then

     

Generators for decompositions of tensor products of modules

If K is a field of finite characteristic p, G is a cyclic group of order q = p ^α , U and W are indecomposable KG-modules, and p ≥ dim U + dim W ? 1, we describe how to find a generator for each of the indecomposable components of the KG-module \({U \otimes W}\).     

Characteristic modules and tensor products over quasi-hereditary algebras

Let A be a monomial quasi-hereditary algebra with a pure strong exact Borel subalgebra B.It is proved that the category of induced good modules over B is contained in the category of good modules over A;that the characteristic module of A is an induced module of that of B via the exact functor-(?)_B A if and only if the induced A-module of an injective B-module remains injective as a B-module.Moreover,it is shown that an exact Borel subalgebra of a basic quasi-hereditary serial algebra is right serial and that the characteristic module of a basic quasi-hereditary serial algebra is exactly the induced module of that of its exact Borel subalgebra.