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39
Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
New branching rules induced by plethysm
 J. Phys A: Math. Gen
, 2006
"... We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi, a symmetric tensor gij = gji an ..."
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We derive group branching laws for formal characters of subgroups Hπ of GL(n) leaving invariant an arbitrary tensor T π of Young symmetry type π where π is an integer partition. The branchings GL(n) ↓ GL(n − 1) , GL(n) ↓ O(n) and GL(2n) ↓ Sp(2n) fixing a vector vi, a symmetric tensor gij = gji and an antisymmetric tensor fij = −fji, respectively, are obtained as special cases. All new branchings are governed by Schur function series obtained from plethysms of the Schur function sπ ≡ {π} by the basic M series of complete symmetric functions and the L = M −1 series of elementary symmetric functions. Our main technical tool is that of Hopf algebras, and our main result is the derivation of a coproduct for any Schur function series obtained by plethysm from another such series. Therefrom one easily obtains πgeneralized NewellLittlewood formulae, and the algebra of the formal group characters of these subgroups is established. Concrete examples and extensive tabulations are displayed for H 1 3, H21, and H3, showing their involved and nontrivial representation theory. The nature of the subgroups is shown to be in general affine, and in some instances non reductive. We discuss the complexity of the coproduct formula and give a graphical notation to cope with it. We also discuss the way in which the group branching laws can be reinterpreted as twisted structures deformed by highly
Generalized descent algebra and construction of irreducible characters of hyperoctahedral groups
, 2006
"... Let (W, S) be a finite Coxeter system and let ℓ: W → N denote the length function. If I ⊂ S, WI =< I> is the standard parabolic subgroup generated by I and XI = {w ∈ W  ∀ s ∈ I, ℓ(ws)> ℓ(w)} is a crosssection of W/WI. Write xI = ∑ ..."
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Let (W, S) be a finite Coxeter system and let ℓ: W → N denote the length function. If I ⊂ S, WI =< I> is the standard parabolic subgroup generated by I and XI = {w ∈ W  ∀ s ∈ I, ℓ(ws)> ℓ(w)} is a crosssection of W/WI. Write xI = ∑
Algebraic aspects of multiple zeta values
 in ”Zeta Functions, Topology and Quantum Physics
, 2005
"... Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding ” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values ca ..."
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Multiple zeta values have been studied by a wide variety of methods. In this article we summarize some of the results about them that can be obtained by an algebraic approach. This involves “coding ” the multiple zeta values by monomials in two noncommuting variables x and y. Multiple zeta values can then be thought of as defining a map ζ: H 0 → R from a graded rational vector space H 0 generated by the “admissible words ” of the noncommutative polynomial algebra Q〈x,y〉. Now H 0 admits two (commutative) products making ζ a homomorphism–the shuffle product and the “harmonic ” product. The latter makes H 0 a subalgebra of the algebra QSym of quasisymmetric functions. We also discuss some results about multiple zeta values that can be stated in terms of derivations and cyclic derivations of Q〈x,y〉, and we define an action of QSym on Q〈x,y 〉 that appears useful. Finally, we apply the algebraic approach to relations of finite partial sums of multiple zeta value series. 1
Sequences of Symmetric Functions of Binomial Type
 Stud. Appl. Math
, 1994
"... We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the results of Umbral Calculus in this context including a Taylor&apo ..."
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We take advantage of the combinatorial interpretations of many sequences of polynomials of binomial type to define a sequence of symmetric functions corresponding to each sequence of polynomials of binomial type. We derive many of the results of Umbral Calculus in this context including a Taylor's expansion and a binomial identity for symmetric functions. Surprisingly, the delta operators for all the sequences of binomial type correspond to the same operator on symmetric functions. Les suites de fonctions sym'etriques de type binomial On s'appuie ici sur les interpr'etations combinatoires de nombreuses suites de polynomes de type binomial pour d'efinir une suite de fonctions sym'etriques associ'ee `a chque suite de polynomes de type binomial. On retrouve dans ce cadre, de nombreaux r'esultats du calcul ombral, en particulier une version de la formule de Taylor et la formule d'identit'e du binome pour les fonctions sym'etriques. On s'aper coit que les op'erateurs differentiels de degr'e un pour toutes les suite de polynomes de type binomial correspondent `a un op'erateur unique sur les fonction sym'etriques. Dedicated to the memory of Rabbi Selig Starr Contents I Linear Sequences 3 1
Hopf algebras and Markov chains: Two examples and a theory
, 2012
"... The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. ..."
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Cited by 10 (5 self)
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The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural “rockbreaking” process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rockbreaking, an explicit description of the quasistationary distribution and sharp rates to absorption follow.