Twisted descent algebras and the Solomon-Tits algebra

The notion of descent algebra of a bialgebra is lifted to the Barratt-Joyal setting of twisted bialgebras. The new twisted descent algebras share many properties with their classical counterparts. For example, there are twisted analogs of classical Lie idempotents and of the peak algebra. Moreover, the universal twisted descent algebra is equipped with two products and a coproduct, and there is a fundamental rule linking all three. This algebra is shown to be naturally related to the geometry of the Coxeter complex of type A.     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

Polynomial identities for the Jordan algebra of a degenerate symmetric bilinear form

Let _Jn$J_n$ be the Jordan algebra of a degenerate symmetric bilinear form. In the first section we classify all possible G  -gradings on _Jn$J_n$ where G   is any group, while in the second part we restrict our attention to a degenerate symmetric bilinear form of rank n−1$n-1$, where n is the dimension of the vector space V   defining _Jn$J_n$. We prove that in this case the algebra _Jn$J_n$ is PI-equivalent to the Jordan algebra of a nondegenerate bilinear form.     

Differential operator specializations of noncommutative symmetric functions

Let K be any unital commutative Q-algebra and z=(z₁,…,zn) commutative or noncommutative free variables. Let t be a formal parameter which commutes with z and elements of K. We denote uniformly by K《z》 and K?t?《z》 the formal power series algebras of z over K and K?t?, respectively. For any α?1, let D[α]《z》 be the unital algebra generated by the differential operators of K《z》 which increase the degree in z by at least α−1 and the group of automorphisms Ft(z)=z−Ht(z) of K?t?《z》 with o(Ht(z))?α and Ht=0(z)=0. First, for any fixed α?1 and , we introduce five sequences of differential operators of K《z》 and show that their generating functions form an NCS (noncommutative symmetric) system [W. Zhao, Noncommutative symmetric systems over associative algebras, J. Pure Appl. Algebra 210 (2) (2007) 363-382] over the differential algebra D[α]《z》. Consequently, by the universal property of the NCS system formed by the generating functions of certain NCSFs (noncommutative symmetric functions) first introduced in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348, MR1327096; see also hep-th/9407124], we obtain a family of Hopf algebra homomorphisms , which are also grading-preserving when Ft satisfies certain conditions. Note that the homomorphisms SFt above can also be viewed as specializations of NCSFs by the differential operators of K《z》. Secondly, we show that, in both commutative and noncommutative cases, this family SFt (with all n?1 and ) of differential operator specializations can distinguish any two different NCSFs. Some connections of the results above with the quasi-symmetric functions [I. Gessel, Multipartite P-partitions and inner products of skew Schur functions, in: Contemp. Math., vol. 34, 1984, pp. 289-301, MR0777705; C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967-982, MR1358493; Richard P. Stanley, Enumerative Combinatorics II, Cambridge University Press, 1999] are also discussed.     

The rational group algebra of a normally monomial group

In this paper, a complete irredundant set of a class of strong Shoda pairs of a finite group G is computed. The algebraic structure of the rational group algebra of a normally monomial group is thus obtained. A necessary and sufficient condition for G to be normally monomial is derived. The main result is also illustrated by computing a complete set of primitive central idempotents and the explicit Wedderburn decomposition of the rational group algebra of some normally monomial groups.     

Spin invariant theory for the symmetric group

We formulate a theory of invariants for the spin symmetric group in some suitable modules which involve the polynomial and exterior algebras. We solve the corresponding graded multiplicity problem in terms of specializations of the Schur Q-functions and a shifted q-hook formula. In addition, we provide a bijective proof for a formula of the principal specialization of the Schur Q-functions.     

Noncommutative symmetric systems over associative algebras

This paper is the first of a sequence of papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] on the (noncommutative symmetric) systems over differential operator algebras in commutative or noncommutative variables [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134]; the systems over the Grossman-Larson Hopf algebras [R. Grossman, R.G. Larson, Hopf-algebraic structure of families of trees, J. Algebra 126 (1) (1989) 184-210. [MR1023294]; L. Foissy, Les algèbres de Hopf des arbres enracinés décorés I, II, Bull. Sci. Math. 126 (3) (2002) 193-239; (4) 249-288. See also math.QA/0105212. [MR1909461]] of labeled rooted trees [W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136]; as well as their connections and applications to the inversion problem [H. Bass, E. Connell, D. Wright, The Jacobian conjecture, reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc. 7 (1982) 287-330. [MR 83k:14028]; A. van den Essen, Polynomial automorphisms and the Jacobian conjecture, in: Progress in Mathematics, vol. 190, Birkhäuser Verlag, Basel, 2000. [MR1790619]] and specializations of NCSFs [W. Zhao, Noncommutative symmetric functions and the inversion problem (submitted for publication). math.CV/0509135; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint]. In this paper, inspired by the seminal work [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348. See also hep-th/9407124. [MR1327096]] on NCSFs (noncommutative symmetric functions), we first formulate the notion of systems over associative Q-algebras. We then prove some results for systems in general; the systems over bialgebras or Hopf algebras; and the universal system formed by the generating functions of certain NCSFs in [I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (2) (1995) 218-348. See also hep-th/9407124. [MR1327096]]. Finally, we review some of the main results that will be proved in the following papers [W. Zhao, Differential operator specializations of noncommutative symmetric functions (submitted for publication). math.CO/0509134; W. Zhao, A system over the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509136; W. Zhao, systems over differential operator algebras and the Grossman-Larson Hopf algebra of labeled rooted trees (submitted for publication). math.CO/0509138. preprint] as some supporting examples for the general discussions given in this paper.     

Free groups and involutions in the unit group of a group algebra

We study the existence of free groups of rank 2 in the group generated by the involutions in the unit group of a group algebra over a non-absolute field.Received: 25 March 2004     

The permutation class group of a finite group

Analogously to the projective class group, the permutation class group of a finite group π can be defined as the group of equivalence classes of direct summands of integral permutation modules modulo permutation modules. It is shown that this group behaves nicely with respect to localization and completion, which then is used to prove that contrary to the projective class group - it is not always a torsion group. More precisely, the rank of the permutation class of group is computed.     

Cyclotomic Solomon algebras

This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type G(r,1,n). As with the Solomon descent algebra, our algebra has a basis given by sums of ‘distinguished’ coset representatives for certain ‘reflection subgroups.’ We explicitly describe the structure constants with respect to this basis and show that they are polynomials in r. This allows us to define a deformation, or q-analogue, of these algebras which depends on a parameter q. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.     

Relations between the Clausen and Kazhdan-Lusztig representations of the symmetric group

We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x_1,1,…,x_n,n] to give a new construction of the Kazhdan-Lusztig representations of S_n. This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x_1,1,…,x_n,n]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young’s natural representations.     

Quantifications des algèbres de Hopf d'arbres plans décorés et lien avec les groupes quantiques

We introduce a functor from the category of braided spaces into the category of braided Hopf algebras which associates to a braided space V a braided Hopf algebra of planar rooted trees . We show that the Nichols algebra of V is a subquotient of . We construct a Hopf pairing between and , generalising one of the results of [Bull. Sci. Math. 126 (2002) 193-239]. When the braiding of c is given by c(v_i⊗v_j)=q_i,jv_j⊗v_i, we obtain a quantification of the Hopf algebras introduced in [Bull. Sci. Math. 126 (2002) 193-239; 126 (2002) 249-288]. When q_i,j=q^a_i,j, with q an indeterminate and (a_i,j)_i,j the Cartan matrix of a semi-simple Lie algebra , then is a subquotient of . In this case, we construct the crossed product of with a torus and then the Drinfel'd quantum double of this Hopf algebra. We show that is a subquotient of .     

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions arise in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.     

The k-orbit reconstruction and the orbit algebra

Let (G, W) be a permutation group on a finite set W = {w ₁,..., w _n}. We consider the natural action of G on the set of all subsets of W. Let h ₀, h ₁,..., h _N be the orbits of this action. For each i, 1 i N, there exists k, 1 k n, such that h _i is a set of k-element subsets of W. In this case h _i is called a symmetrized k-orbit of the group (G, W) or simply a k-orbit. With a k-orbit h _i we associate a multiset H(h _i ) = % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeoaaa!3690!\[\langle \]h _i ⁽¹⁾, h _i ⁽²⁾,..., h _i ^(k)% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJepaaa!36A1!\[\rangle \] of its (k – 1)-suborbits. Orbits h _i and h _j are called equivalent if H(h _i ) = H(h _j ). An orbit is reconstructible if it is equivalent to itself only. The paper concerns the k-orbit reconstruction problem and its connections with different problems in combinatorics. The technique developed is based on the notion of orbit and co-orbit algebras associated with a given permutation group (G, W).     

A polynomial realization of the Hopf algebra of uniform block permutations

We provide a polynomial realization of the Hopf algebra UBP of uniform block permutations defined by Orellana and Aguiar (2008) [11]. We describe an embedding of the dual of the Hopf algebra WQSym into UBP, and as a consequence, obtain a polynomial realization of it.     

Noncommutative symmetric functions and an amazing matrix

We present a simple way to derive the results of Diaconis and Fulman [P. Diaconis, J. Fulman, Foulkes characters, Eulerian idempotents, and an amazing matrix, arXiv:1102.5159] in terms of noncommutative symmetric functions.     

Schur–Weyl dualities for symmetric inverse semigroups

We obtain Schur–Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations.     

Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks

We develop a number of statistical aspects of symmetric groups (mostly dealing with the distribution of cycles in various subsets of Sn), asymptotic properties of (ordinary) characters of symmetric groups, and estimates for the multiplicities of root number functions of these groups. As main applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, a finiteness result for the number of Fuchsian presentations of such a group (resolving a long-standing problem of Roger Lyndon), as well as a proof of a well-known conjecture of Roichman concerning the mixing time of random walks on symmetric groups.     

Graded Brauer tree algebras

In this paper we construct non-negative gradings on a basic Brauer tree algebra A_Γ corresponding to an arbitrary Brauer tree Γ of type (m,e). We do this by transferring gradings via derived equivalence from a basic Brauer tree algebra A_S, whose tree is a star with the exceptional vertex in the middle, to A_Γ. The grading on A_S comes from the tight grading given by the radical filtration. To transfer gradings via derived equivalence we use tilting complexes constructed by taking Green’s walk around Γ (cf. Schaps and Zakay-Illouz (2001) [17]). By computing endomorphism rings of these tilting complexes we get graded algebras.We also compute , the group of outer automorphisms that fix the isomorphism classes of simple A_Γ-modules, where Γ is an arbitrary Brauer tree, and we prove that there is unique grading on A_Γ up to graded Morita equivalence and rescaling.     

On the derived algebra of a centraliser

Let g be a classical Lie algebra, e∈g a nilpotent element and ge⊂g the centraliser of e. We prove that ge=[ge,ge] if and only if e is rigid. It is also shown that if e∈[ge,ge], then the nilpotent radical of ge coincides with [ge(1),ge], where ge(1)⊂ge is an eigenspace of a characteristic of e corresponding to the eigenvalue 1.