Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

Bergeron, Nantel; Reutenauer, Christophe; Rosas, Mercedes; Zabrocki, Mike

doi:10.4153/cjm-2008-013-4

Cited by 48 publications

(102 citation statements)

References 12 publications

Supporting

Mentioning

101

Contrasting

Order By: Relevance

“…, P l ) ∈ X . 6,5) Then (P ) = P and (−1) (P ) = −(−1) (P ) for all P ∈ X , that is, P → P defines is a sign-reversing pairing of the summands of c R,S . This implies that c R,S = 0, hence that e A is Δ-primitive.…”

Section: Corollary 44 Let a ∈ Fin And Suppose Thatmentioning

confidence: 98%

See 1 more Smart Citation

The module structure of the Solomon–Tits algebra of the symmetric group

Schocker¹

2006

Journal of Algebra

View full text Add to dashboard Cite

Let (W, S) be a finite Coxeter system. Tits defined an associative product on the set Σ of simplices of the associated Coxeter complex. The corresponding semigroup algebra is the Solomon-Tits algebra of W . It contains the Solomon algebra of W as the algebra of invariants with respect to the natural action of W on Σ. For the symmetric group S n , there is a 1-1 correspondence between Σ and the set of all set compositions (or ordered set partitions) of {1, . . . , n}. The product on Σ has a simple combinatorial description in terms of set compositions. We study in detail the representation theory of the Solomon-Tits algebra of S n over an arbitrary field, and show how our results relate to the corresponding results on the Solomon algebra of S n . This includes the construction of irreducible and principal indecomposable modules, a description of the Cartan invariants, of the Ext-quiver, and of the descending Loewy series. Our approach builds on a (twisted) Hopf algebra structure on the direct sum of all Solomon-Tits algebras.

show abstract

Section: Corollary 44 Let a ∈ Fin And Suppose Thatmentioning

confidence: 98%

“…From a different point of view, this is the attempt to pursue the study of symmetric and quasi-symmetric functions in non-commuting variables initiated in [30] and developed a great deal further in [5,6,24].…”

Section: Introductionmentioning

confidence: 98%

The module structure of the Solomon–Tits algebra of the symmetric group

Schocker¹

2006

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…There is a natural Hopf algebra structure on this graded algebra (which we review in Sect. 5.2), and following [7,14,21,30], we call NCSym the Hopf algebra of symmetric functions in noncommuting variables. NCSym is one of the several noncommutative analogs of the more familiar and much-studied Hopf algebra of symmetric functions, which we denote Sym.…”

Section: Classical Bases Of Symmetric Functionsmentioning

confidence: 99%

Strong forms of linearization for Hopf monoids in species

Marberg¹

2015

J Algebr Comb

View full text Add to dashboard Cite

A vector species is a functor from the category of finite sets with bijections to vector spaces; informally, one can view this as a sequence of $S_n$-modules. A Hopf monoid (in the category of vector species) consists of a vector species with unit, counit, product, and coproduct morphisms satisfying several compatibility conditions, analogous to a graded Hopf algebra. We say that a Hopf monoid is strongly linearized if it has a "basis" preserved by its product and coproduct in a certain sense. We prove several equivalent characterizations of this property, and show that any strongly linearized Hopf monoid which is commutative and cocommutative possesses four bases which one can view as analogues of the classical bases of the algebra of symmetric functions. There are natural functors which turn Hopf monoids into graded Hopf algebras, and applying these functors to strongly linearized Hopf monoids produces several notable families of Hopf algebras. For example, in this way we give a simple unified construction of the Hopf algebras of superclass functions attached to the maximal unipotent subgroups of three families of classical Chevalley groups.Comment: 35 pages; v2: corrected some typos, fixed attribution for Theorem 5.4.4; v3: some corrections, slight revisions, added references; v4: updated references, numbering of results modified to conform with published version, final versio

show abstract

“…It is known that the natural coproduct of WSym (given as usual by the ordered sum of alphabets) is cocommutative [4] and that WSym is free over connected set partitions. The same argument as in Theorem 2.5 shows that ΠQSym * is free over the same graded set, hence that ΠQSym is indeed isomorphic to WSym * .…”

Section: Interpretation Of H(v ) Thementioning

confidence: 99%

Commutative combinatorial Hopf algebras

2007

View full text Add to dashboard Cite

Abstract. We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.

show abstract

Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables

Cited by 48 publications

References 12 publications

The module structure of the Solomon–Tits algebra of the symmetric group

The module structure of the Solomon–Tits algebra of the symmetric group

Strong forms of linearization for Hopf monoids in species

Commutative combinatorial Hopf algebras

Contact Info

Product

Resources

About