On the Descent Algebra of Type DD

Here we give a combinatorial interpretation of Solomon's rule for multiplication in the descent algebra of Weyl groups of type $D$, $\Sigma D_n$. From here we show that $\Sigma D_n$ is a homomorphic image of the descent algebra of the hyperoctahedral group, $\Sigma B_{n-2}$.Comment: 7 page

The partition algebra and the Kronecker product (Extended Abstract)

We propose a new approach to study the Kronecker coefficients by using the Schur–Weyl duality between the symmetric group and the partition algebra

Induction and Restriction of Kazhdan–Lusztig Cells

AbstractBarbasch and Vogan gave a beautiful rule for restricting and inducing Kazhdan–Lusztig representations of Weyl groups. In this paper we show that this rule implies and generalizes the Littlewood–Richardson rule for decomposing outer products of representations of the symmetric groups. A new recursive rule for computing characters of arbitrary Coxeter groups follows. Another application is a generalization of Garsia–Remmel's algorithm for decomposing certain tensor products of symmetric groups representations

Homomorphisms between Solomon's descent algebras

In a previous paper (see A. Garsia and C. Reutenauer (Adv. in Math. 77, 1989, 189–262)), we have studied algebraic properties of the descent algebras Σn, and shown how these are related to the canonical decomposition of the free Lie algebra corresponding to a version of the Poincaré-Birkhoff-Witt theorem. In the present paper, we study homomorphisms between these algebras Σn. The existence of these homomorphisms was suggested by properties of some directed graphs that we constructed in the previous paper (reference above) describing the structure of the descent algebras. More precisely, examination of the graphs suggested the existence of homomorphisms Σn→Σn−s and Σn→Σn+s. We were then able to construct, for any s (0<s<n), a surjective homomorphism Δs: Σn→Σn−s and an embedding Γs:Σn−s→Σn, which reflects these observations. The homomorphisms Δs may also be defined as derivations of the free associative algebra Q〈t1,t2,…> which sends ti on ti−s, if one identifies the basis element D⊆S of Σn with some word (coding S) on the alphabet T={t1, t2,…}. We show that this mapping is indeed a homomorphism, using the combinatorial description of the multiplication table of Σn given in the previous paper (reference above)

The Hopf structure of symmetric group characters as symmetric functions

In arXiv:1605.06672 the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.Comment: arXiv admin note: text overlap with arXiv:1605.0667