RICERCA ITALIANA     

Polynomial identities and combinatorial methods

Università degli Studi di Palermo

Abstract

The primary objective of the research project is the study of the polynomial identities satisfied by an algebra over a field of characteristic zero via combinatorial methods pertaining to the representation theory of the symmetric and general linear groups, to algebraic combinatorics, to invariant theory. This approach has been very fruitful in the past, and is based on the theory of varieties developed by Kemer. In this theory a fundamental role is played by the superalgebras and their identities. This approach was introduced and developed mainly by Berele, Drensky, Formanek, Kemer, Procesi, Razmyslov and Regev.
We plan to develop further the full strength of the superalgebraic version of the Capelli method of "auxiliary variables", in order to find decomposition results for some classes of "plethystic algebras"; we also plan to contribute to the "constructive invariant theory" of classical groups, with main reference to the extension of the method of transvectants, to the extension of the symbolic method to the ring of polynomial functions on arbitrary Schur/Weyl modules and to the representation theory of finite reflection groups on polynomial algebras.
This program will also involve detailed studies of specific combinatorial structures as well as the related Hopf algebras; among them, we mention the RKS-correspondence, permutation (involution) statistics and the theory of non crossing partitions.
In the theory of varieties one attaches to an algebra some numerical invariants such as the sequence of codimensions, the sequence of cocharacters and the sequence of colengths and we plan to contribute to the theory through the study of their asymptotics. In the case of superalgebras (but also of algebras with involution or of algebras graded by a finite group G) one defines corresponding finer invariants (determined through the representation theory of the wreath products G wr S_n). From a comparison of these with the classical invariants, one can obtain a better understanding of the ordinary polynomial identities. <<<

Principal Investigator

Antonino Giambruno Università degli Studi di PALERMO

Research Objectives

The final objectives of this research project are the following.

Computation of the exponent of a variety: to extend the classification of the varieties determined by Capelli polynomials and Amitsur polynomials to the minimal varieties. In this setting we shall try to find a closer relation between a given variety and the minimal varieties with the same exponent.
Classification of the varieties of polynomial growth: to extend the recently determined classification of the varieties of linear growth and of the subvarieties of the variety generated by the Grassmann algebra or the algebra of 2 x 2 upper triangular matrices. We shall also work on the conjecture stating that that from a certain value of the polynomial growth (&gt;6?), the list of algebras characterizing the varieties is no more finite.
Growth of the Lie codimensions and of the proper codimensions: to determine if the limit of the sequences {c_n^p(A)^(1/n)} and {c_n^L(A)^(1/n)} exist and are integers. We plan to solve this problem for finitely generated algebras and in the general case for the proper codimensions.
Nonassociative algebras and exponential growth: to study the finite dimensional nonassociative algebras and to try to classify their exponential growth (is it an integer?). For simple Jordan algebras does the exponent exist and equals the dimension of the algebra?
Superalgebras and algebras with involution: to compare the invariants determined through the representation theory of the hyperoctahedral group of the supercodimensions or the *-codimensions with the classical ones.
Supervarieties and Grassmann envelope: to work on the problem of determining a generating algebra of a supervariety which is related to the Grassmann algebra and to the finite dimensional Z_2 x Z_2- graded algebras.
Combinatorics of Schur functions: by extending previous results for the algebra M_1,1(K), to find an explicit formula for the multiplicities m(?) in the trace cocharacters of the algebra M_2,1(K) through the study of the Kronecker products of Schur functions.
Invariants of n x n matrices: to find generators and relations of minimal degree for the algebra Cnd of the invariants of GLn acting via simultaneous conjugation on d n x n matrices, for all d=3.
G-graduations, functional identities: to study the graded polynomial identities of the verbally prime algebras and to determine all possible graduations through finite abelian groups on special algebras. To study the functional identities with automorphisms and generalized derivations of a ring.
Letterplace superalgebras: to extend part of this theory to a characteristic-free setting, namely, to obtain universal forms up to a filtration (composition series) of some complete decomposition results that hold in characteristic zero. To exploit the connections between the theory of letterplace superalgebras and the general representation theory of general linear Lie superalgebras in order to study the link between the combinatorial description and their description as Kac modules ("typical modules") or quotients of Kac modules ("atypical modules").
Algebras of invariants: to extend the combinatorial approach to the problem of generating minimal finite systems of generators for the algebra of SL-invariants of n-ary forms, n an arbitrary positive integer, to the study of skew-symmetric tensors and, more generally, to explore the possibility of adapting these techniques to other classes of rational representations of linearly reductive algebraic groups.
Umbral method: to develop a version of the umbral method that unifies different generalizations in order to obtain combinatorial descriptions of the irreducible submodules of super(skew)symmetric algebras on superalgebraic Schur/Weyl modules.
Actions of reflection groups: Let W be a finite reflection group acting on a vector space of dimension n. Considering the dual action of W one can define in a natural way several actions on the ring of polynomials in nk variables. To study the Hilbert series of the associated algebras of invariants and covariants by generalizing the results recently obtained in the special case of the polynomial ring in 2n variables.
Skew partitions and symmetric generatine functions: One considers product and coproduct of posets, obtaining a Hopf algebra. Among labelled posets there is the class of those associated to the Ferrers diagram of skew partitions. Such posets form a Hopf subalgebra. The generating function of these posets is symmetric. Stanley conjectured that the converse is true, that is: if the generating function of a poset P is symmetric, then P necessarily is of the special kind associated to a skew partition. This conjecture is still open. To study and to try to solve Stanley's conjecture, using the Hopf algebraic structure of quasi-symmetric functions.
Distributions of involutions: to study the Eulerian distribution on the involutions with no fixed points and on the self evacuated involutions, namely, involutions corresponding to standard Young tableaux which are fixed under the Schützenberger map. More precisely, the present approach will give an explicit formula for the number j2n,k of self evacuated involutions on 2n objects with k descents.
Umbral calculus: to apply the umbral method to Schur functions by simplifying by symbolic techniques some of the most useful formulas like the Littlewood-Richardson rule. Moreover we intend to extend the umbral notion of cumulant to the free case, with the aim to study Kerov formula expressing certain characters of the symmetric group in terms of free cumulants of the associated Young diagrams.
Grassmann-Clifford geometric calculus: to study further the connections among Cayley-Grassmann algebras (in the sense of Rota), geometric Clifford algebras (in the sense of Hestenes) and covariant theory of skew-symmetric tensors; we hope to develop a version of the so-called Cliffordization process in terms of superpolarizations of virtual variables. <<<

First Results

Taking into account the final targets of the research project, the expected results represent a significant contribution to the combinatorial theory of algebras with polynomial identities and to various aspects of the algebraic combinatorics and its applications. The expected results can be described as follows.

1) In the asymptotic computation of the codimension sequences of an associative algebra we shall try to obtain further information on the varieties relatyed to the exponent. Given a T-ideal of the free algebra by generators, we shall try to determine the exponent of the corresponding variety in some significant cases. Since an asymptotic equality between the sequence of codimensions of a variety of such type with some special generators (standard identity, Capelli identity, Amitsur identity) and the codimension sequence of an algebra of matrices was determined earlier, we intend to extend this classification to the minimal varieties.

2) Starting from the explicit classification, up to PI-equivalence, of the algebras whose sequence of codimensions is linearly bounded, and from the classification of the subvarieties of the variety generated by the Grassmann algebra or the algebra of 2 x 2 upper triangular matrices, we expect to extend these results to higher growth. In particular we shall work at the conjecture stating that for a suitable value of the growth (&gt;6) the list of algebras characterizing the corresponding varieties is no longer finite.

3) Two more numerical sequences related to the polynomial identities of a given algebra are of interest: the sequence of proper codimensions c_n^p(A) and the sequence of Lie codimensions c_n^L(A). Such sequences are both bounded from above by the usual codimension sequence. In this setting an open problem is to determine if limits of the sequences {c_n^p(A)^(1/n)} and {c_n^L(A)^(1/n)} exist and are integers. It is our intention to study this problem for finitely generated algebras and in general for the sequence of proper codimensions.

4) In case of non necessarily associative algebras, for any real number x greater than 1, a variety of exponential growth equal to x has been constructed. Moreover it has been shown that the finite dimensional algebras cannot have intermediate growth. In this framework, we wish to study the finite dimensional algebras trying to classify their exponential growth (is it an integer?). In the special case of simple Jordan algebras, does the exponent exists and is it equal to the dimension of the algebra?

5) An interesting refinement of the theory of algebras with polynomial identities is obtained when the algebra has a further structure of superalgebra or of algebra with involution *, by studying the corresponding superidentities or *-identities. This situation arises when the given algebra has an automorphism or an antiautomorphism of order 2. The exponential growth of a *-variety generated by a finite dimensional algebra has been determined in an explicit way. A similar generalization has also been carried out for the supervarieties in case they are generated by a finitely generated PI-algebra. In both cases we shall compare the newly determined invariants with the classical ones.

6) In the description of varieties of algebras the fundamental result of Kemer claims that any proper variety is generated by the Grassmann envelope of a finite dimensional superalgebra. This result stating the relations of the identities with the superidentities of a finite dimensional superalgebra allows to develop the theory significantly in the last decade. The principle obstruction of the development of the theory of superalgebras is the absence of an analogous result for the supervarieties. We plan to work on this problem by trying to determine a generating algebra for a supervariety which is related to the Grassmann algebra and to the finite dimensional Z_2xZ_2-graded algebras.

7) Another problem we wish to study is that of a possible classification of the supervarieties and of the *-varieties with given polynomial growth of the codimensions. This study was started with the classification of the supervarieties and of the *-varieties of at most linear growth. This result allowed to describe the possible linear sequences that can appear as sequences of the supercodimensions or *-codimensions of an algebra.

8) When studying the problem of determining the identities of matrices there is an approach based on the combinatorics of the Schur functions. The theory of trace identities developed independently by Procesi and Razmyslov is a powerful tool for the study of the identities satisfied by the algebra of rxr matrices. From this theory and from its hook generalization due to Regev, it follows that the computation of the multiplicities m(?) in the trace cocharacters of the matrix algebra M_r(K), where ?= (?1,…, ?k) is a partition of n with at most r^2 non zero parts, can be reduced to the study of certain Kronecker products of Schur functions. Such research has been recently developed by Carini in cooperation with Remmel, Whitehead and Regev. Recently Carbonara, Carini and Remmel determined, for any r, the behaviour of m(?) when ?2…, ?r is fixed and ?1 sufficiently large. We shall try to get more information on these multiplicities. We also plan to extend the results obtained by Remmel for the algebra M_1,1(K) to the case of the algebra M_2,1(K).

9) The study of the algebra of n x n generic matrices introduced by Procesi is stricty related to the invariant theory of n x n matrices. One studies the algebra Cnd of invariants of the general linear group GLn acting by simultaneous conjugation on d matrices of size n. General results of invariant theory of classical groups imply that the algebra Cnd is finitely generated. The theory of PI-algebras provides upper bounds for the generating sets of the algebra Cnd. A description of the defining relations of Cnd is given by the Razmyslov-Procesi theory. Explicit minimal sets of generators of Cnd and the defining relations between them are found in few cases only. Benanti e Drensky have recently determined the minimal degree of the set of defining relations of C3d for any d=3 and all relations of minimal degree. We plan to study the algebra C4d, find the generators and the defining relations of minimal degree for any d=3.

10) If G is a finite abelian group, it is well known that a G-graduation on an algebra A over an algebraically closed field of characteristic zero, determines G as a group of automorphisms of A and viceversa. Therefore the study of the G-graded identities is reduced to that of the G-dentities (identities with automorphisms). The essential tool is the representation theory of the wreath product G wr S_n. In this setting it is also natural to try to determine all possible graduations, through finite abelian groups, on certain algebras of special interest.

11) Other topics that we plan to study include the relation between the polynomial identities satisfied by an algebra and the group identities satisfied by its group of units. A conjecture of Hartley for group algebras KG states that if the group G is torsion, K is any field and the group of units of KG satisfies a group identity, then the group algebra KG satisfies a polynomial identity. It turns out that the polynomial identities that can appear are very special. In this framework we believe it is possible to get a more general classification for symmetric units under an involution induced by the group G.

12) The theory of letterplace superalgebras over fields of characteristic zero, regarded as bimodules with respect to the action of a pair of general linear Lie superalgebras, is a rich and fairly general one; it simplifies and encompasses several classical theories, such as, for example, the representation theory of the general linear and symmetric groups, the Berele-Regev representation theory on spaces of Z_2-graded tensors. We plan to extend part of this theory to a characteristic-free setting, namely, to obtain universal forms up to a filtration (composition series) of some complete decomposition results that hold in characteristic zero.

13) The irreducible submodules of letterplace superalgebras are Schur/Weyl modules (also called "covariant modules") and, using the superalgebraic version of the method of virtual variables, they admit an effective combinatorial description. In order to exploit the connections between the theory of letterplace superalgebras and the general representation theory of general linear Lie superalgebras we will study the link between the combinatorial description and their description as Kac modules ("typical modules") or quotients of Kac modules ("atypical modules").

14) The theme of finding algorithms for Hilbert's finitness theorem is a relevant theme of current mathematical research and a lot of work has been done in this direction; the major methodology is the modern theory of Groebner bases, in combination with tools borrowed from commutative algebra and algebraic geometry. On the other hand, algorithms were obtained by Gordan in the case of binary forms. In recent times, the combinatorial approach has been extended to the problem of generating minimal finite systems of generators for the algebra of SL-invariants of n-ary forms, n an arbitrary positive integer. This approach is based on a generalization of the so-called "electro-chemical method" of Sylvester for binary forms and involves a detailed analysis of the combinatorics of "transvectants", in the case of n-ary forms. We plan to develop an analogous study for skew-symmetric tensors and, more generally, to explore the possibility of adapting these techniques to other classes of rational representations of linearly reductive algebraic groups.

15) The method of Grosshans, Rota and Stein was generalized to certain classes of plethystic algebras (super(skew)symmetric algebras on spaces of homogeneous supersymmetric tensors) thereby obtaining, for example, a combinatorial construction of all their irriducible submodules with respect to the action of general linear Lie superalgebras. In recent times, Keet has obtained a generalization of the umbral method to algebras of polynomial functions on arbitrary Schur/Weyl modules. We plan to develop a version of the umbral method that unifies these two different generalizations in order to obtain combinatorial descriptions of the irreducible submodules of super(skew)symmetric algebras on superalgebraic Schur/Weyl modules.

16) Let W be a finite reflection group acting on a vector space of dimension n. Considering the dual action of W one can define in a natural way several actions on the ring of polynomials in nk variables. One can consider the Hilbert series of the associated invariant and coinvariant algebras. We propose to study these Hilbert series generalizing the results obtained in the special case of the polynomial ring in 2n variables. Such generalization should involve the property of Cohen-Macaulayness of these invariant algebras and a deep analysis of the decomposition into irreducible modules of the tensor product between two irreducible representations of a finite reflection group.

17) Malvenuto considered a product and coproduct of posets, obtaining a Hopf algebra. Among labelled posets there is the class of those associated to the Ferrers diagram of skew partitions. Such posets form a Hopf subalgebra. The generating function of these posets is symmetric. Stanley conjectured that the converse is true, that is: if the generating function of a poset P is symmentric, then P necessarily is of the special kind associated to a skew partition. We want to study and try to solve Stanley's conjecture, using the Hopf algebraic structure of quasi-symmetric functions.

18) The distribution of the descent statistic on the set of involutions of the symmetric group Sn has been deeply investigated in recent years by several authors, who studied the combinatorial properties of the generating polynomial In(x) of the descent distribution. This distribution was conjectured to be log-concave by F. Brenti. The key tool is a formula that expresses the number in,k of involutions on n objects with k descents in terms of the well known sequence an,s counting semistandard tableaux with n cells on s symbols. This approach gives a further and simpler proof of the symmetry of the coefficients of the polynomial In(x). We aim to apply the same approach to the solution of other enumerative problems concerning some subsets of involutions. More precisely, the present approach will give an explicit formula for the number j2n,k of self evacuated involutions on 2n objects with k descents.

19) The umbral interpretation of cumulants has open a new route for a general theory for k-statistics and their generalizations. This subject is closely related to symmetric functions and the umbral interpretations of k-statistics has produced surprising computational results on classical bases of the algebra of symmetric functions. Following this ideas, we are going to apply this umbral method to Schur functions simplifying by symbolic techniques some of the more useful formulas like the Littlewood-Richardson rule. Moreover we intend to extend the umbral notion of cumulant to the free case, with the aim to study the Kerov formula that express some characters of the symmetric group in terms of free cumulants of Young diagrams associates.

20) We plan to study the functional identities involving automorphisms and generalized derivations of a ring, aiming at a complete description for the latter. Subsequently we intend to analyze the structure of the subrings and subgroups generated by suitable functional polynomials which are not identities of the ring. In particular, we want to discriminate in which cases such subrings (subgroups, respectively) contain ideals (or Lie ideals, respectively) of the given ring.

21) We plan to further analyze the class of algebras with involution recently introduced by Di Vincenzo and La Scala. These are subalgebras of block triangular matrix algebras with involution obtained by flipping a matrix through the secondary diagonal. It has been conjectured that those algebras with involution generate precisely the varieties which are minimal with respect to the *-exponent. We intend to enlarge the set of cases in which this conjecture has been positively verified. Next step is to investigate the structure of the ideal of *-polynomial identities of the verbally prime superalgebras M_{p,q}(G). Here M_{p,q}(G) is endowed with the natural involution induced by a pair of superinvolutions, defined on the matrix superalgebra M_{p+q}(F) with entries in the field F and on the Grassmann algebra G, respectively. <<<

Timescale

24 months

National and international background

The use of the representation theory of the symmetric group for studying the T-ideals of the free associative algebra was systematically made since the early 70's; this method was developed in characteristic zero by Regev, Kemer, Procesi, Razmyslov, Amitsur, Berele, Drensky and others.
The theory of Lie algebras with polynomial identities was developed by the Moscow school by Bahturin, Zaicev, Mishchenko, Petrogradski, Razmyslov and others (Y. Bahturin, A. A. Mikhalev, V. Petrogradsky, M. Zaicev, Infinite Dimesional Lie Superalgebras, Walter de Gruyter, Berlin, 1992).

The study of the asymptotic behavior of the sequence of codimensions for associative algebras was started by Regev by proving that for a PI-algebra the sequence of codimensions is exponentially bounded (Existence of identities in A x B, Israel J. Math. 11 (1972), 131-152). The theorem asserting that every proper variety has integral exponential growth and its explicit computation is due to Giambruno and Zaicev (On codimension growth of finitely generated associative algebras, Adv. Math. 140 (1998), 145-155; Exponential codimension growth of PI-algebras: an exact estimate, Adv. Math., 142 (1999), 221-243). The computation of the exponent and the methods introduced have allowed to compute the exponential growth of polynomials. This study was started by Berele and Regev in [ Exponential growth for codimensions of some p.i. algebras. J. Algebra 241 (2001), 118-145 ]. The study of the varieties of polynomial growth was started by Kemer in [T-ideals with polynomial growth of the codimensions are Specht, Sib. Math. J. 19 (1978), 37-48 ]. Drensky and Regev, among others, have contributed to this area. In the last 3 years the classification of the varieties of linear growth (A. Giambruno, and D. La Mattina, PI-algebras with slow codimension growth, J. Algebra 284 (2005), no. 1, 371-391) and of the subvarieties of the varieities of the Grassmann algebra and of the algebra of 2x2 upper triangular matrices have been obtained (D. La Mattina, Varieties of almost polynomial growth: classifying their subvarieties, Manuscripta Math. 123, 185-203).

The study of the polynomial identities for a superalgebra or for an algebra with involution via the representation theory of the hyperoctahedral group was started by Giambruno and Regev (Wreath products and PI-algebras, J. Pure Applied Algebra 35 (1985) 133-149).
The study of the asymptotic behavior of the Lie codimensions of a Lie algebra has been carried out by several authors. The most significant contributions are due to Bahturin, Zaicev, Mishchenko, Petrogradski, Razmyslov (see Y. Bahturin, A. A. Mikhalev, V. Petrogradsky, M. Zaicev, Infinite Dimesional Lie Superalgebras, Walter de Gruyter, Berlin, 1992). In general for non associative algebras it was proved by Giambruno, Mishchenko and Zaicev that the exponential growth of an algebra can be any real number greater than 1. In case of not necessarily associative finite dimensional algebras it was proved in [ A. Giambruno, S. Mishchenko and M. Zaicev, Algebras with intermediate growth of the codimensions, Adv. in Applied Math. 37 (2006), no. 3, 360-377 ] that they cannot have intermediate growth of the codimensions.

The computation of the sequence of cocharacters for the 2x2 matrices is due to Drensky (Polynomial identities for 2x2 matrices, Acta Appl. Math. 21 (1990), No.1/2, 137-161) and Procesi (Computing with 2x2 matrices, J. Algebra 87 (1984), 342-359). The introduction and the study of the trace identities for the nxn matrices is due independently to Procesi (The invariant theory of nxn matrices, Adv.in Math.19 (1976), 306-381) and Razmyslov (Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR, Izv. 8 (1974), 727-760). In [Sign Trace Identities, Linear andMultilinear Algebra, 1987, Vol 21, pp.1-28], Regev applies the representation theory of the general linear Lie superalgebra to (hook)generalize the theory of trace identities. From [The invariant theory of nxn matrices, Adv.in Math.19 (1976), 306-381], [Trace identities of full matrix algebras over a field of characteristic zero, Math. USSR, Izv. 8 (1974), 727-760] and [Sign Trace Identities, Linear andMultilinear Algebra, 1987, Vol 21, pp.1-28] it follows that the computation of the multiplicities in the trace (or sign trace) cocharacters (and so in most of the ordinary cocharacters) can be reduced to the study of certain Kronecker products of Schur functions.

The study of the algebra of n x n generic matrices is stricly related to the invariant theory of n x n matrices. The theory of PI-algebras provides upper bounds for the generating sets of the algebra Cnd of the invariants of the general linear group GLn acting by simultaneous conjugation on d matrices of size n. A description of the defining relations of Cnd is given by the Razmyslov-Procesi theory (C.Procesi, The invariant theory of nxn matrices, Adv.in Math.19 (1976), 306-381; Y. Razmyslov, Identities of Algebras and Their Representations, Transl. Amer. Math. Monogr., vol. 138, Amer. Math. Soc., Providence RI, 1994). In the last years generators and relations of C3d have been determined (F. Benanti, V. Drensky, Defining relations of noncommutative trace algebra of two 3 x 3 matrices, Adv. Appl. Math. 37 (2006), 162-182).

The study of the graded identities of an algebra through the representation theory of the wreath products GwrS_n was started by Giambruno and Regev in [Wreath products and PI-algebras, J. Pure Applied Algebra 35 (1985), 133-149]. In [A. Giambruno, S. Mishchenko and M. Zaicev, Group actions and asymptotic behavior of graded polynomial identities, J. London Math. Soc. (2) 66 (2002), 295-312], the authors characterized all varietes of G-graded algebras, G a finite abelian group, having polynomial growth.

A comprehensive treatment of the theory of letterplace superalgebras as bimodules with respect to a pair of general linear Lie superalgebras, as well as of its applications, can be found in the recent work: A. Brini, Combinatorics, Superalgebras, Invariant Theory and Representation Theory, Seminaire Lotharingien de Combinatoire, Vol 55 (2007), 114pp. (electronic publication).

The state of the art of constructive invariant theory, as well as of its links with the theory of Groebner bases and methods borrowed from algebraic geometry, can be found in the recent volume: Derksen H., Kemper G., Computational invariant theory, Encyclopaedia of Mathematical Sciences. Invariant Theory and Algebraic Transformation Groups. 130 (1) (2002), Springer. For a modern version of the method of transvectants in the binary case, see: Olver P.J., Classical Invariant Theory, London Mathematical Society Student Texts 44 (1999), Cambridge University Press. This method has been recently extended to the n-ary case in [Brini A., Regonati F., Teolis A., Combinatorics, Transvectants and Superalgebras. An elementary combinatorial appoach to Hilbert's Finiteness Theorem, Adv. in Math. 37 (3) (2006), 287 – 308].

The superalgebraic version of the symbolic method was introduced by Grosshans, Rota and Stein (Invariant Theory and Superalgebras, Amer. Math. Soc., Providence, RI, 1987) in order to study joint covariants of symmetric and skew-symmetric tensors. This method was then generalized by Brini, Huang and Teolis (The umbral symbolic method for supersymmetric tensors, Adv. in Math. 96 (1992), 123-193) and Grosshans (The Symbolic Method and Representation Theory, Adv. Math. 98 (1993), 113-142) in order to study some classes of plethystic superalgebras. Recently, A. Keet (preprint) developed a different version of the symbolic method that applies to the study of the coordinate ring of arbitrary Schur modules.

In the last years there has been much interest in the study of invariant and coinvariant algebras with respect to the action of a (finite) reflection group on the ring of polynomials in several variables.
This study has shed more light on the combinatorics of Weyl groups with particular interest on "descents" and "major indices". Moreover, the representation theory of these groups led to the definition of new modules indexed by purely combinatorial objects such as partitions and related things, whose decompositions into irreducible modules involve in a crucial way other combinatorial objects such as Young tableaux and several statistics defined on them. Among the more recent contributions, we mention R.Adin, F.Brenti, Y.Roichman (Descent Representations and multivariate statistics, Trans. Amer. Math. Soc., 357 (2005), 3051-3082), R.Biagioli, F.Caselli (Invariant algebras and major indices for classical Weyl groups, Proc. London Math. Soc. 88 (2004), 603-631), H.Barcelo, V.Reiner, D.Stanton (Bimahonian Distributions, preprint, math/0703479).

A well-known theme in algebraic combinatorics, initiated by Rota, is that many discrete structures are endowed in a natural way with a product and a coproduct: their enumeration and classification give rise to an associated Hopf algebraic structure, which encodes the assembling and disassembling of these structures. Examples are the algebra of Malvenuto-Reutenauer of permutations, the algebra of Loday-Ronco of planar binary trees, the algebra of trees of Connes-Kreimer, the algebra of Reutenauer-Poirier of Young tableaux, the algebra of symmetric functions and quasi-symmetric functions and many more. These aspects matured in a recent work of Aguiar, Bergeron e Sottile (Combinatorial Hopf algebras and generalized Dehn-Sommerville equations, Compositio Mathematica 142 (2006), 1-30),where they give a natural morphism between Hopf algebras of combinatorial objects and the Hopf algebra of quasi-symmetric functions QSym.

A comprehensive treatment of the geometric calculus based on the theory of geometric Clifford algebras can be found in the volume: G. Sommer (ed.) (Geometric computing with Clifford algebras : theoretical foundations and applications in computer vision and robotics, Springer, 2001) (see also D. Hestenes, G. Sobczyk, (Clifford Algebra to Geometric Calculus. A unified language for mathematics and phisics, Reidel, 1992)). A standard reference for the theory of Cayley-Grassmann algebras is the paper M. Barnabei, A. Brini, G.-C. Rota (On the exterior calculus of invariant theory, J. Algebra 96 (1985), 120-160). Relevant extensions of these ideas have been obtained by H. Crapo and W. Schmitt (The Whitney algebra of a matroid, J. Combin. Theory Ser. A 91 (2000), no. 1-2, 215-263). <<<