Noncommutative Pieri Operators on Posets

Bergeron, Nantel; Mykytiuk, Stefan; Sottile, Frank; Willigenburg, Stephanie van

doi:10.1006/jcta.2000.3090

Cited by 51 publications

(91 citation statements)

References 26 publications

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“…Here, our approach is to study properties of equivalence relations on G = n≥0 G n and to determine when they give rise to algebraic structures. Many of the algebraic objects obtained in this way are G-colouring of the structures in the literature, such as the peak algebras [1,9,21] or the Loday-Ronco Hopf algebra of trees [17]. Our work is the unifying generalization of a series of results starting in [5] and continuing in [6,14,26].…”

Section: Introductionmentioning

confidence: 91%

“…The peak algebraP is freely generated byp (n) for n odd [9,Theorem 5.4]. Then the freeness follows from Corollary 3.8, and the functorial property follows from Corollary 3.9.…”

Section: Proofmentioning

confidence: 99%

“…We give some applications of this theory in Section 4. In particular, we describe a G-coloured peak Hopf algebra (a G-colouring of the peak Hopf algebra from [9]), and a G-coloured Loday-Ronco Hopf algebra (a G-colouring of the Loday-Ronco Hopf algebra [17]). To our knowledge, these two G-coloured algebras are new.…”

Section: Introductionmentioning

confidence: 99%

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Coloured peak algebras and Hopf algebras

Bergeron

Hohlweg

2006

J Algebr Comb

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For G a finite abelian group, we study the properties of general equivalence relations on G n = G n S n , the wreath product of G with the symmetric group S n , also known as the G-coloured symmetric group. We show that under certain conditions, some equivalence relations give rise to subalgebras of kG n as well as graded connected Hopf subalgebras of n≥o kG n . In particular we construct a G-coloured peak subalgebra of the Mantaci-Reutenauer algebra (or G-coloured descent algebra). We show that the direct sum of the G-coloured peak algebras is a Hopf algebra. We also have similar results for a G-colouring of the Loday-Ronco Hopf algebras of planar binary trees. For many of the equivalence relations under study, we obtain a functor from the category of finite abelian groups to the category of graded connected Hopf algebras. We end our investigation by describing a Hopf endomorphism of the G-coloured descent Hopf algebra whose image is the G-coloured peak Hopf algebra. We outline a theory of combinatorial G-coloured Hopf algebra for which the G-coloured quasi-symmetric Hopf algebra and the graded dual to the G-coloured peak Hopf algebra are central objects.

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Section: Introductionmentioning

confidence: 91%

“…The peak algebraP is freely generated byp (n) for n odd [9,Theorem 5.4]. Then the freeness follows from Corollary 3.8, and the functorial property follows from Corollary 3.9.…”

Section: Proofmentioning

confidence: 99%

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Coloured peak algebras and Hopf algebras

Bergeron

Hohlweg

2006

J Algebr Comb

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“…In [10], these coproducts on QSym and A, respectively, are shown to extend to coproducts on Π and A E , and they proved [10,Theorem 5.4…”

Section: Peak Functions and Eulerian Posetsmentioning

confidence: 88%

“…The main result for our purposes with respect to the sublagebra Π is due to Bergeron, Mykytiuk, Sottile and van Willigenburg [10].…”

Section: Peak Functions and Eulerian Posetsmentioning

confidence: 98%

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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