Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

King, Ronald C.; Welsh, Trevor A.; Willigenburg, Stephanie J. van

doi:10.1007/s10801-007-0113-0

Cited by 14 publications

(23 citation statements)

References 14 publications

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“…It is therefore natural to consider the expansions of other symmetric functions in the basis of Schur functions. are Schur positive, and such questions have been the subject of much recent work, such as [1,4,9,10,11,12,15,17]. It is well-known that these questions are currently intractable when stated in anything close to full generality.…”

Section: Introductionmentioning

confidence: 99%

Positivity results on ribbon Schur function differences

McNamara

Willigenburg

2009

European Journal of Combinatorics

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There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur-positivity is essentially a product of two chains.

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Section: Introductionmentioning

confidence: 99%

Positivity results on ribbon Schur function differences

McNamara

Willigenburg

2009

European Journal of Combinatorics

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“…For each long intermediate row of the ribbon, there is a unique such tableau where that row is filled with a single 1 followed by 2's. from Chain (15) shows that this is not the case. In Section 6 we explain what is going on here.…”

Section: Large Ribbons and Short End Rowsmentioning

confidence: 94%

Necessary Conditions for Schur-Maximality

Tom

Willigenburg

2018

Electron. J. Combin.

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McNamara and Pylyavskyy conjectured precisely which connected skew shapes are maximal in the Schur-positivity order, which says that B ≤ s A if s A − s B is Schurpositive. Towards this, McNamara and van Willigenburg proved that it suffices to study equitable ribbons, namely ribbons whose row lengths are all of length a or (a + 1) for a ≥ 2. In this paper we confirm the conjecture of McNamara and Pylyavskyy in all cases where the comparable equitable ribbons form a chain. We also confirm a conjecture of McNamara and van Willigenburg regarding which equitable ribbons in general are minimal.Additionally, we establish two sufficient conditions for the difference of two ribbons to be Schur-positive, which manifest as diagrammatic operations on ribbons. We also deduce two necessary conditions for the difference of two equitable ribbons to be Schur-positive that rely on rows of length a being at the end, or on rows of length (a + 1) being evenly distributed. Contents2010 Mathematics Subject Classification. Primary 05E05; Secondary 05E10, 06A05, 06A06, 20C30.

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“…It has three outer corners (marked with •) in positions (7, 1), (5,3), and (3,5). We can take η 7,1 = (5, 5), η 5,3 = (5, 5, 2, 2, 2, 2, 2), and η 3,5 = (5,4,4,4,2,2). The partition in Figure 15 (b) is not corner-symmetric.…”

Section: Definition 312mentioning

confidence: 99%

“…See for example [1,2,3,4,5,7,13]. These expressions can also be interpreted as differences of skew Schur functions which have been studied in [11,12].…”

Section: Introductionmentioning

confidence: 99%

Schur-Positivity in a Square

Ballantine

Orellana

2014

Electron. J. Combin.

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Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schurpositivity of a family of symmetric functions. Given a partition ν, we denote by ν c its complement in a square partition (m m ). We conjecture a Schur-positivity criterion for symmetric functions of the form s µ ′ sµc − s ν ′ sνc , where ν is a partition of weight |µ| − 1 contained in µ and the complement of µ is taken in the same square partition as the complement of ν. We prove the conjecture in many cases.

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Schur positivity of skew Schur function differences and applications to ribbons and Schubert classes

Cited by 14 publications

References 14 publications

Positivity results on ribbon Schur function differences

Positivity results on ribbon Schur function differences

Necessary Conditions for Schur-Maximality

Schur-Positivity in a Square

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