Graded Characters of Modules Supported in the Closure of a Nilpotent Conjugacy Class

Shimozono, Mark; Weyman, Jerzy

doi:10.1006/eujc.1999.0344

Cited by 55 publications

(143 citation statements)

References 20 publications

(31 reference statements)

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“…The positive expansion of Macdonald polynomials indexed by 2-bounded partitions (equivalently, partitions with ℓ(λ) ≤ 2) into atoms of level 2 is thus conjecturally the one given in [7,21] (and to [18], since [19] proves the operators are related to the functions studied in [18]). …”

Section: Case K = 2 and K =mentioning

confidence: 99%

Tableau atoms and a new Macdonald positivity conjecture

Lapointe¹,

Lascoux²,

Morse³

2003

Duke Math. J.

132

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Let Λ be the space of symmetric functions and V k be the subspace spanned by the modified Schur functions {S λ [X/(1 − t)]} λ1≤k . We introduce a new family of symmetric polynomials, {A, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials A (k) λ [X; t] form a basis for V k and that the Macdonald polynomials indexed by partitions whose first part is not larger than k expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the A (k) λ [X; t] relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the A (k) λ [X; t] seem to play the same role for V k as the Schur functions do for Λ. In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients.

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Section: Case K = 2 and K =mentioning

confidence: 99%

Tableau atoms and a new Macdonald positivity conjecture

Lapointe¹,

Lascoux²,

Morse³

2003

Duke Math. J.

132

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“…Generalized Kostka polynomials [26,33,35,36,37,38] are q-analogues of the tensor product multiplicity…”

Section: Introductionmentioning

confidence: 99%

Fermionic Formulas for Level-Restricted Generalized Kostka Polynomials and Coset Branching Functions

Schilling

Shimozono

2001

Communications in Mathematical Physics

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Abstract. Level-restricted paths play an important rôle in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood-Richardson tableaux and rigged configurations is well-behaved with respect to level-restriction and give an explicit characterization of level-restricted rigged configurations. As a consequence a new general fermionic formula for the level-restricted generalized Kostka polynomial is obtained. Some coset branching functions of type A are computed by taking limits of these fermionic formulas.

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“…There is a conjectured combinatorial interpretation for the expansion of H λ ( * ) [X; t] (see [148]) if λ ( * ) is a sequence of partitions such that the concatenation of the partitions in λ ( * ) is a partition (i.e. if for each of the adjacent partitions in λ ( * ) we have λ…”

Section: A Symmetric Function Operator Definitionmentioning

confidence: 99%

“…The operators B λ from Equation (3.6) were defined by Shimozono and Zabrocki in [149] as a tool for understanding the generalized Kostka polynomials (also known as parabolic Kostka coefficients) that were studied by Shimozono and Weyman [148]. The combinatorial interpretation for the Schur expansion of a composition of these operators when the indexing partitions concatenate to a partition in terms of katabolizable tableaux is still an open problem.…”

Section: Notes On Referencesmentioning

confidence: 99%

“…An explicit rule for the kbranching formula with a t is conjectured in [82,Conjecture 1]. [148] that H λ ( * ) [X; t] is Schur positive whenever the concatenation of all of the partitions of λ ( * ) form a partition (which happens for all k-splits of partitions). Conjecture 3.10 is likely to follow from this conjecture and hence so would the property thatÃ …”

Section: The K-schur Function Is Schur Positivementioning

confidence: 99%

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