Abstract. The Fomin-Kirillov algebra E n is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define E G to be the subalgebra of E n generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, E

Abstract. Recently Blasiak gave a combinatorial rule for the Kronecker coefficient g λµν when µ is a hook shape by defining a set of colored Yamanouchi tableaux with cardinality g λµν in terms of a process called conversion. We give a characterization of colored Yamanouchi tableaux that does not rely on conversion, which leads to a simpler formulation and proof of the Kronecker rule for one hook shape.

Abstract. For permutations v, w ∈ S n , Macdonald defines the skew divided difference operators ∂ w/v as the unique linear operators satisfying ∂ w (P Q) = v v(∂ w/v P ) · ∂ v Q for all polynomials P and Q. We prove that ∂ w/v has a positive expression in terms of divided difference operators ∂ ij for i < j. In fact, we prove that the analogous result holds in the Fomin-Kirillov algebra E n , which settles a conjecture of Kirillov.

Abstract. We prove that the dimension of the Specht module of a forest G is the same as the normalized volume of the matching polytope of G. We also associate to G a symmetric function s G (analogous to the Schur symmetric function s λ for a partition λ) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of G. We then use this to define notions of standard and semistandard tableaux for forests.