Recent developments in algebraic combinatorics

Stanley, Richard P.

doi:10.1007/bf02803505

Cited by 22 publications

(4 citation statements)

References 33 publications

(38 reference statements)

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“…This motivation is centred around the main result of [15]. A nice introduction, with emphasis on the context and the importance of Postnikov's result can be found in [20]. Here, however, we merely extract from these two references the minimum amount of background necessary to show how Postnikov's work ties together cylindric skew Schur functions and an important open problem in Quantum Schubert Calculus.…”

Section: Cylindric Skew Schur Functionsmentioning

confidence: 99%

“…While we do not claim that this paragraph is sufficient to give a firm understanding of C λ,d µν , we do claim that it is clear from this geometric definition that C λ,d µν ≥ 0. No algebraic or combinatorial proof of this inequality is known and, as stated in [20], it is a fundamental open problem to find such a proof.…”

Section: Cylindric Skew Schur Functionsmentioning

confidence: 99%

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Cylindric skew Schur functions

McNamara

2006

Advances in Mathematics

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Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining these motivations, we study cylindric skew Schur functions from the point of view of Schur-positivity. Using a result of I. Gessel and C. Krattenthaler, we generalise a formula of A. Bertram, I. Ciocan-Fontanine and W. Fulton, thus giving an expansion of an arbitrary cylindric skew Schur function in terms of skew Schur functions. While we show that no non-trivial cylindric skew Schur functions are Schur-positive, we conjecture that this can be reconciled using the new concept of cylindric Schur-positivity.

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Section: Cylindric Skew Schur Functionsmentioning

confidence: 99%

Section: Cylindric Skew Schur Functionsmentioning

confidence: 99%

Cylindric skew Schur functions

McNamara

2006

Advances in Mathematics

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“…Undoubtedly, Richard's best known such work is Chapter 7 of Enumerative Combinatorics [Sta99]. Other examples are [Sta83], which was particularly helpful prior to the advent of Appendix 2 of [Sta99], and each of [Sta00, Sta03,Sta04] includes at least one section about symmetric functions.…”

Section: Theory and Application Of Plane Partitionsmentioning

confidence: 99%

The contributions of Stanley to the fabric of symmetric and quasisymmetric functions

Billey¹,

McNamara²

2016

The Mathematical Legacy of Richard P. Stanley

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We weave together a tale of two rings, SYM and QSYM, following one gold thread spun by Richard Stanley. The lesson we learn from this tale is that "Combinatorial objects like to be counted by quasisymmetric functions." arXiv:1505.01115v2 [math.CO] 28 May 2015 1 Richard recalls writing [Sta71b] as a graduate student, some time before the summer of 1970. See [Sta14] for more details on the timing of Richard's graduation.

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“…This was proved by Stanley in [47] and [48] for simplicial and then all rational polytopes using the cohomology of toric varieties, and extended to all polytopes by Karu [37], by means of the theory of combinatorial intersection cohomology. See [21] or [53] for a discussion of this combinatorial cohomology theory.…”

Section: Eulerian Posets and The Cd-indexmentioning

confidence: 99%

Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups

Billera

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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Abstract. We discuss the enumeration theory for flags in Eulerian partially ordered sets, emphasizing the two main geometric and algebraic examples, face posets of convex polytopes and regular CW -spheres, and Bruhat intervals in Coxeter groups. We review the two algebraic approaches to flag enumeration -one essentially as a quotient of the algebra of noncommutative symmetric functions and the other as a subalgebra of the algebra of quasisymmetric functions -and their relation via duality of Hopf algebras. One result is a direct expression for the Kazhdan-Lusztig polynomial of a Bruhat interval in terms of a new invariant, the complete cd-index. Finally, we summarize the theory of combinatorial Hopf algebras, which gives a unifying framework for the quasisymmetric generating functions developed here.Mathematics Subject Classification (2010). Primary 06A11; Secondary 05E05, 16T30, 20F55, 52B11.

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Recent developments in algebraic combinatorics

Cited by 22 publications

References 33 publications

Cylindric skew Schur functions

Cylindric skew Schur functions

The contributions of Stanley to the fabric of symmetric and quasisymmetric functions

Flag Enumeration in Polytopes, Eulerian Partially Ordered Sets and Coxeter Groups

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