Opposite Littlewood-Richardson sequences and their matrix realizations

Azenhas, Olga

doi:10.1016/0024-3795(93)00324-s

Cited by 6 publications

(15 citation statements)

References 3 publications

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“…For instance, (4, 3, 1, 1) is the conjugate of a = (4, 2, 2, 1). Given two partitions a and c, we write a ⊆ c to mean a i c i , for all i. Graphically, this means that the Young diagram of a is contained in the Young diagram of c. A skew diagram c/a is obtained by removing the smaller diagram of a from the diagram of c. For example, if a = (4, 2, 2, 1) and c = (5,4,4,3,2), the following shows the skew diagram c/a:…”

Section: Young Tableaux and Wordsmentioning

confidence: 99%

“…is a (skew) tableau of type ((4, 2, 2, 1), (4,3,2), (5,4,4,3,2)), with indexing sets A sequence (J 1 , . .…”

Section: Young Tableaux and Wordsmentioning

confidence: 99%

“…Definition 2.3 [5]. A word w over the alphabet [t] is said a Yamanouchi word [13] if any right factor v of w satisfies |v| 1 |v| 2 · · · |v| t .…”

Section: Young Tableaux and Wordsmentioning

confidence: 99%

“…Then, (T, H σ ) is an admissible pair if and only if w(T) ≡ H σ .Proof. Let F 1 , F 2 be the indexing sets of T. From[2] and[5], (T, H σ ) is an admissible pair if and only if w(F 1 , F 2 ) is a Yamanouchi word, when σ = id, and w(F 1 , F 2 ) is a dual Yamanouchi word, when σ = s 1 . Therefore, the result follows from Proposition 4.3.…”

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confidence: 99%

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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: Young Tableaux and Wordsmentioning

confidence: 99%

“…is a (skew) tableau of type ((4, 2, 2, 1), (4,3,2), (5,4,4,3,2)), with indexing sets A sequence (J 1 , . .…”

Section: Young Tableaux and Wordsmentioning

confidence: 99%

“…Definition 2.3 [5]. A word w over the alphabet [t] is said a Yamanouchi word [13] if any right factor v of w satisfies |v| 1 |v| 2 · · · |v| t .…”

Section: Young Tableaux and Wordsmentioning

confidence: 99%

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confidence: 99%

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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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“…The first version of the 2-by-2 block matrix problem over a field F [given just above (2)] gives rise to a result that we state below as Result 4, which parallels Result 3, where p is now a monic, irreducible polynomial over F. Note that, in the field case, we cannot escape the obvious degree-dimension consistency condition: if S and T are square F-matrices, having orders s and t, and having, respectively, p α 1 , . .…”

mentioning

confidence: 99%