Hopf Algebra Of Symmetric Functions Research Articles

Pattern groups and a poset based Hopf monoid

The supercharacter theory of algebra groups gave us a representation theoretic realization of the Hopf algebra of symmetric functions in noncommuting variables. The underlying representation theoretic framework comes equipped with two canonical bases, one of which was completely new in terms of symmetric functions. This paper simultaneously generalizes this Hopf structure by considering a larger class of groups while also restricting the representation theory to a more combinatorially tractable one. Using the normal lattice supercharacter theory of pattern groups, we not only gain a third canonical basis, but also are able to compute numerous structure constants in the corresponding Hopf monoid, including coproducts and antipodes for the new bases.

The Hopf algebra of skew shapes, torsion sheaves on [formula omitted], and ideals in Hall algebras of monoid representations

We study ideals in Hall algebras of monoid representations on pointed sets corresponding to certain conditions on the representations. These conditions include the property that the monoid act via partial permutations, that the representation possess a compatible grading, and conditions on the support of the module. Quotients by these ideals lead to combinatorial Hopf algebras which can be interpreted as Hall algebras of certain sub-categories of modules. In the case of the free commutative monoid on n generators, we obtain a co-commutative Hopf algebra structure on n-dimensional skew shapes, whose underlying associative product amounts to a “stacking” operation on the skew shapes. The primitive elements of this Hopf algebra correspond to connected skew shapes, and form a graded Lie algebra by anti-symmetrizing the associative product. We interpret this Hopf algebra as the Hall algebra of a certain category of coherent torsion sheaves on A/F1n supported at the origin, where F1 denotes the field of one element. This Hopf algebra may be viewed as an n-dimensional generalization of the Hopf algebra of symmetric functions, which corresponds to the case n=1.

A Hopf Algebraic Approach to Schur Function Identities

Using cocommutativity of the Hopf algebra of symmetric functions, certain skew Schur functions are proved to be equal. Some of these skew Schur function identities are new.

Hopf algebras, distributive (Laplace) pairings and hash products: a unified approach to tensor product decompositions of group characters

We show for bicommutative graded connected Hopf algebras that a certain distributive (Laplace) subgroup of the convolution monoid of 2-cochains parameterizes certain well behaved Hopf algebra deformations. Using the Laplace group, or its Frobenius subgroup, we define higher derived hash products, and develop a general theory to study their main properties. Applying our results to the (universal) bicommutative graded connected Hopf algebra of symmetric functions, we show that classical tensor product and character decompositions, such as those for the general linear group, mixed co- and contravariant or rational characters, orthogonal and symplectic group characters, Thibon and reduced symmetric group characters, are special cases of higher derived hash products. In the appendix we discuss a relation to formal group laws.

General Dyson–Schwinger Equations and Systems

We classify combinatorial Dyson–Schwinger equations giving a Hopf subalgebra of the Hopf algebra of Feynman graphs of the considered Quantum Field Theory. We first treat single equations with an arbitrary (eventually infinite) number of insertion operators. We distinguish two cases; in the first one, the Hopf subalgebra generated by the solution is isomorphic to the Faa di Bruno Hopf algebra or to the Hopf algebra of symmetric functions; in the second case, we obtain the dual of the enveloping algebra of a particular associative algebra (seen as a Lie algebra). We also treat systems with an arbitrary finite number of equations, with an arbitrary number of insertion operators, with at least one of degree 1 in each equation.

A SUPERCHARACTER TABLE DECOMPOSITION VIA POWER-SUM SYMMETRIC FUNCTIONS

We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over 𝔽q, into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q-1]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group Sngiven by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular, we compute the determinant of the supercharacter table.

Noncommutative irreducible characters of the symmetric group and noncommutative Schur functions

In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we show that in the Hopf algebra of noncommutative symmetric functions, NSym, of which Sym is a quotient, the recently discovered basis of noncommutative Schur functions exhibits that every noncommutative Schur function is isomorphic to a noncommutative irreducible character of a symmetric group when working in noncommutative character theory. We simultaneously show that a second basis of NSym consisting of Young noncommutative Schur functions also satises that every element is isomorphic to a noncommutative irreducible character of a symmetric group.

The Hopf algebra structure of the character rings of classical groups

The character ring of covariant irreducible tensor representations of the general linear group admits a Hopf algebra structure isomorphic to the Hopf algebra of symmetric functions. Here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions. We show that and also admit natural Hopf algebra structures that are isomorphic to that of , and hence to . The isomorphisms are determined explicitly, along with the specification of standard bases for and analogous to those used for . A major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic Schur–Hall scalar products. Significantly, the adjoint with respect to multiplication no longer coincides, as it does in the case, with a Foulkes derivative or skew operation. The adjoint and Foulkes derivative now require separate definitions, and their properties are explored here in the orthogonal and symplectic cases. Moreover, the Hopf algebras and are not self-dual. The dual Hopf algebras and are identified. Finally, the Hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified.

The primitives and antipode in the Hopf algebra of symmetric functions in noncommuting variables

We identify a collection of primitive elements generating the Hopf algebra NCSym of symmetric functions in noncommuting variables and give a combinatorial formula for the antipode.

The Hopf algebra of uniform block permutations

We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Rosas, and Sagan. These two embeddings correspond to the factorization of a uniform block permutation as a product of an invertible element and an idempotent one.

Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This paper attempts to survey the ongoing investigations in this topic as dictated by the knowledge and interests of its author. There are many open questions that are discussed.

Plethystic Hopf algebras.

The notion of a plethystic algebra associated with a Hopf algebra endowed with a suitable bilinear form is defined. A special case is the Hopf algebra of symmetric functions.