Matrix realizations of littlewood—richardson sequences

Azenhas, Olga; Sá, Eduardo Marques de

doi:10.1080/03081089008818015

Cited by 15 publications

(22 citation statements)

References 6 publications

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“…. , m t ) is by decreasing order, T is a Littlewood-Richardson tableau [1][2][3]. Now, for each partition a and n × n unimodular matrix U , let T (a,M) (U ) be the set of all sequences of matrices, as above, with (m 1 , .…”

Section: Introductionmentioning

confidence: 99%

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Azenhas¹,

Mamede²

2005

Linear Algebra and its Applications

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Let M be the set of all rearrangements of t fixed integers in {1, . . . , n}. We consider those Young tableaux T, of weight (m 1 , . . . , m t ) in M, arising from a sequence of products of matrices over a local principal ideal domain, with maximal ideal (p), a , a U(pI m 1 ⊕ I n−m 1 ), a U 2 k=1 (pI m k ⊕ I n−m k ),where a is an n × n nonsingular diagonal matrix, with invariant partition a, and U is an n × n unimodular matrix. Given a partition a and an n × n unimodular matrix U , we consider the set T (a,M) (U ) of all sequences of matrices, as above, with (m 1 , . . . , m t ) running over M. The symmetric group acts on T (a,M) (U ) by place permutations of the tuples in M. When t = 2, 3, the action of the symmetric group on the set of Young tableaux, having the set T (a,M) (U ) as matrix realization, is described by a decomposition of the indexing sets of ୋ Work supported by CMUC/FCT. the Littlewood-Richardson tableau in T (a,M) (U ), afforded by the matrix U . This description, in cases t = 2, 3, gives necessary and sufficient conditions for the existence of an unimodular matrix U such that T (a,M) (U ) is a matrix realization of a set of Young tableaux, with given shape c/a and weight running over M. If H is the tableau arising from the sequence of matrices, above, when a = 0, it is shown that the words of the tableaux T and H are Knuth equivalent. The relationship between this action of the symmetric group and the one described by A. Lascoux and M.P. Schutzenberger [Noncommutative structures in algebra and geometric combinatorics

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Section: Introductionmentioning

confidence: 99%

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Azenhas¹,

Mamede²

2005

Linear Algebra and its Applications

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“…For the matrix product problem, say, this means that, if we solve the problem when A and B are viewed as matrices over R p , i.e., if we characterize the power-of-p elementary divisors of a matrix product, for each individual prime p, then we get a solution to the general problem, by just merging together chains of prime power elementary divisors. This is done by R. Thompson in [25] (see also [3], for products of possibly singular matrices). Localization and primary decompositions are well-understood techniques both in abstract commutative algebra and module theory (e.g.…”

Section: Invariant Factorsmentioning

confidence: 99%

Group representations and matrix spectral problems^∗

Santana

Queiró

Sá

1999

Linear and Multilinear Algebra

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“…Suppose T is a nilpotent operator on H. Obviously the number λ 1 = max{h(x, {0}) : x ∈ H} can be attained for some v 1 …”

Section: Lemma 11 Let T ∈ L(h) Be Of Type λ Thenmentioning

confidence: 99%

“…w + e c,δ 1 . In both cases v ∈ N (T δ 1 ) and T e c,δ 1 ∈ M. This and the induction assumption implies (5) and statements (1)- (4) for j = 2, .…”

Section: Definition 21 Let T ∈ L(h) T ∈ L(h )mentioning

confidence: 99%

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Invariant subspaces of nilpotent operators and LR-sequences

Müller²

1999

Integr equ oper theory

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Invariant subspaces of nilpotent operators and LR-sequencesWing Suet Li and Vladimír Müller 0. INTRODUCTION It is well-known that an operator in a finite-dimensional vector space is uniquely determined up to a similarity by its Jordan model.Given an operator T , two invariant subspaces M and N of T are said to be similar if there exists an invertible operator X such that XT = T X and XM = N .Clearly this is an equivalence relation in the lattice of invariant subspaces of T .Our paper was originally motivated by the following question raised by The aim of the present paper is to study systematically invariant subspaces of finite-dimensional nilpotent operators. The paper is organized as follows. In Section 1 we recall some of the needed facts from linear algebra and set up the necessary notation. In Section 2 we study the case of cyclic T M (the Jordan model of T M consists of a single block). Among others we show that in this case the Jordan model of T |M determines the similarity orbit of M. In Section 3 we study relations between invariant subspaces and various types of LR-sequences. We provide another constructive solution of Klein's result, and at the same time, a better picture of the relation between Littlewood-Richardson sequences and the structure of the associated operator T .In Section 4 we obtain a convexity result about Littlewood-Richardson sequences. The paper concludes in Section 5 with a study of different equivalence classes of invariant suspaces, which envile some interesting properties of the lattice of invariant subspaces of a nilpotent operator on a finite-dimensional vector space.

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Matrix realizations of littlewood—richardson sequences

Cited by 15 publications

References 6 publications

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Group representations and matrix spectral problems^∗

Invariant subspaces of nilpotent operators and LR-sequences

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Matrix realizations of littlewood—richardson sequences

Cited by 15 publications

References 6 publications

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Action of the symmetric group on sets of skew-tableaux with prescribed matrix realization

Group representations and matrix spectral problems∗

Invariant subspaces of nilpotent operators and LR-sequences

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Product

Resources

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Group representations and matrix spectral problems^∗