1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetric groups 4. Representations of wreath products 5. Applications to combinatories and representation theory 6. Modular representations 7. Representation theory of Sn over an arbitrary field 8. Representations of general linear groups Appendices Index.

The representation theory of the symmetric group

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

Publisher Summary This chapter explores that the Lie algebra of Lie polynomials is the free Lie algebra. Lie polynomials appeared at the end of the 19th century and the beginning of the 20th century in the work of Campbell, Baker and Hausdorff on exponential mapping in a Lie group, which lead to the Campbell–Baker–Hausdorff formula. Around 1930, Witt showed that the Lie algebra of Lie polynomials is the free Lie algebra, and that its enveloping algebra is the associative algebra of noncommutative polynomials. The Poincare–Birkhoff–Witt theorem is proved, and shows that the free Lie algebra is related to the lower central series of a free group. The full linear group acts on Lie polynomials, and the symmetric group acts on those which are multilinear. The Lie representation of the symmetric group is induced from any faithful representation of a subgroup generated by a full cycle. Automorphism of a free Lie algebra are always tame, and are characterized by a Jacobian condition. The full linear group acts on Lie polynomials, and the symmetric group acts on those which are multilinear. The descent algebra is dual to the ring of quasi-symmetric functions which is, therefore, a free commutative algebra.

Free Lie Algebras

Groupes et algébres de lie

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

https://hal.archives-ouvertes.fr/hal-00017721/document

Noncommutative Symmetrical Functions

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47).

Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics).

In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … 

[↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu.

 [1]: #xref-corresp-1-1

https://www.pnas.org/content/pnas/111/27/9676.full.pdf

Cluster algebras

We present a fast algorithm for computing the global crystal basis of the basic
 
$$U_q (\widehat{\mathfrak{s}\mathfrak{l}}_n )$$

 -module. This algorithm is based on combinatorial techniques which have been developed for dealing with modular representations of symmetric groups, and more generally with representations of Hecke algebras of typeA at roots of unity. We conjecture that, upon specializationq→1, our algorithm computes the decomposition matrices of all Hecke algebras at an
th root of 1.

/pdf/hecke-algebras-at-roots-of-unity-and-crystal-bases-of-4ghb3cy6mo.pdf

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.

/pdf/rigid-modules-over-preprojective-algebras-2j2seujyxu.pdf

Rigid modules over preprojective algebras

Let C be the category of finite-dimensional representations of a quantum affine algebra U q ( g ∧ ) of simply laced type. We introduce certain monoidal subcategories C l ( l ∈ N ) of C , and we study their Grothendieck rings using cluster algebras.

/pdf/cluster-algebras-and-quantum-affine-algebras-4hu9mkcyf1.pdf

Bernard Leclerc

Papers

Noncommutative Symmetrical Functions

Cluster algebras

Hecke algebras at roots of unity and crystal bases of quantum affine algebras

Rigid modules over preprojective algebras

Cluster algebras and quantum affine algebras