Subgroup lattices and symmetric functions

Butler, Lynne M.

doi:10.1090/memo/0539

Cited by 87 publications

(134 citation statements)

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“…(For the precise information, see [8].) On the other hand, it was shown in [2] that there exists close relationship between the subgroup lattice of a finite abelian p-group G of type λ = (λ 1 , . .…”

Section: Ordered Partitions Of a Multisetmentioning

confidence: 99%

“…In [2], Butler introduced an inversion-like statistic on Tab(λ, k) to show that the rankselected Möbius invariant arising from the subgroup lattice of a finite abelian p-group of type λ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(λ, k) and OP k λ…”

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confidence: 99%

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Inversion-like and Major-like Statistics of an Ordered Partition of a Multiset

Choi¹

2016

Kyungpook mathematical journal

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Abstract. Given a partition λ = (λ1, λ2, . . . , λ l ) of a positive integer n, let Tab(λ, k) be the set of all tabloids of shape λ whose weights range over the set of all k-compositions of n and OP k λ rev the set of all ordered partitions into k blocks of the multiset {1In [2], Butler introduced an inversion-like statistic on Tab(λ, k) to show that the rankselected Möbius invariant arising from the subgroup lattice of a finite abelian p-group of type λ has nonnegative coefficients as a polynomial in p. In this paper, we introduce an inversion-like statistic on the set of ordered partitions of a multiset and construct an inversion-preserving bijection between Tab(λ, k) and OP k λ. When k = 2, we also introduce a major-like statistic on Tab(λ, 2) and study its connection to the inversion statistic due to Butler.

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Section: Ordered Partitions Of a Multisetmentioning

confidence: 99%

mentioning

confidence: 99%

Inversion-like and Major-like Statistics of an Ordered Partition of a Multiset

Choi¹

2016

Kyungpook mathematical journal

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“…where α λ (µ; p) is the number of subgroups of type µ in a finite abelian p-group of type λ, [2][3][4]20].…”

Section: The Konvalinka-lauve Formulas and Their Q-analoguesmentioning

confidence: 99%

Remarks on the paper “Skew Pieri rules for Hall–Littlewood functions” by Konvalinka and Lauve

Warnaar

2013

J Algebr Comb

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Abstract. In a recent paper Konvalinka and Lauve proved several skew Pieri rules for Hall-Littlewood polynomials. In this note we show that q-analogues of these rules are encoded in a q-binomial theorem for Macdonald polynomials due to Lascoux and the author. The Konvalinka-Lauve formulas and their q-analoguesWe refer the reader to [14] for definitions concerning Hall-Littlewood and Macdonald polynomials.Let P λ/µ = P λ/µ (X; t) and Q λ/µ = Q λ/µ (X; t) be the skew Hall-Littlewood polynomials, e r = P (1 r ) the rth elementary symmetric function, h r the rth complete symmetric function and q r = Q (r) . Then the ordinary Pieri formulas for HallLittlewood polynomials are given by [14] P µ e r = λ vs λ/µ (t)P λ (1.1a)where the sums on the right are over partitions λ such that |λ| = |µ| + r. The Pieri coefficient vs λ/µ (t) is given by [14, p. 215, (3.2so that vs λ/µ (t) is zero unless µ ⊆ λ with λ− µ a vertical r-strip. Similarly, hs λ/µ (t) vanishes unless µ ⊆ λ with λ − µ a horizontal r-strip, in which case [14, p. 218, (3.10)]To express the skew Pieri formulas, Konvalinka and Lauve [8] (see also [7]) introduced a third Pieri coefficientwhere n(λ/µ) := i≥1. Note that sk λ/µ (t) = 0 if µ ⊆ λ.

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“…Remark A consequence of this theorem is that the Kostka numbers K µλ (1) are the multiplicities of the (finite-dimensional) sl(n)-modules in the Demazure modules E w ( ).…”

Section: Theorem 8 the K µλ (Q) Have Positive Coefficientsmentioning

confidence: 99%

“…The main advantage of our approach is its simplicity and its explanation of the connection with Macdonald polynomials. Nonnegativity and positivity of Kostka polynomials have already been proven by Lascoux-Schützenberger [9], Butler [1], Lusztig [10]. The connection between the branching rule and Kostka polynomials was explored in [5].…”

Section: Introductionmentioning

confidence: 99%