We introduce the immersion poset (P(n),⩽I) on partitions, defined by λ⩽Iμ if and only if sμ(x1,…,xN)-sλ(x1,…,xN) is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of GLN(C) form an immersion pair, as defined by Prasad and Raghunathan [7]. We develop injections SSYT(λ,ν)↪SSYT(μ,ν) on semistandard Young tableaux given constraints on the shape of λ, and present results on immersion relations among hook and two column partitions. The standard immersion poset (P(n),⩽std) is a refinement of the immersion poset, defined by λ⩽stdμ if and only if λ⩽Dμ in dominance order and fλ⩽fμ, where fν is the number of standard Young tableaux of shape ν. We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12].

In 1961, Solomon proved that the sum of all the entries in the character table of a finite group does not exceed the cardinality of the group. We state a different and incomparable property here – this sum is at most twice the sum of dimensions of the irreducible characters. We establish the validity of this property for all finite irreducible Coxeter groups. The main tool we use is that the sum of a column in the character table of such a group is given by the number of square roots of the corresponding conjugacy class representative. We then show that the asymptotics of character table sums is the same as the number of involutions in symmetric, hyperoctahedral and demihyperoctahedral groups. Finally, we derive generating functions for the character table sums for these latter groups as well as generalized symmetric groups as infinite products of continued fractions.

Let G be a finite group and p be a prime number dividing the order of G. An irreducible character χ of G is called a quasi p-Steinberg character if χ(g) is nonzero for every p-regular element g in G. In this paper, we classify the quasi p-Steinberg characters of complex reflection groups G(r,q,n) and exceptional complex reflection groups. In particular, we obtain this classification for Weyl groups of type Bn and type Dn.

We introduce the multiset partition algebra MPk(ξ) over the polynomial ring F[ξ], where F is a field of characteristic 0 and k is a positive integer. When ξ is specialized to a positive integer n, we establish the Schur—Weyl duality between the actions of resulting algebra MPk(n) and the symmetric group Sn on Symk(Fn). The construction of MPk(ξ) generalizes to any vector λ of non-negative integers yielding the algebra MPλ(ξ) over F[ξ] so that there is Schur—Weyl duality between the actions of MPλ(n) and Sn on Symλ(Fn). We find the generating function for the multiplicity of each irreducible representation of Sn in Symλ(Fn), as λ varies, in terms of a plethysm of Schur functions. As consequences we obtain an indexing set for the irreducible representations of MPk(n) and the generating function for the multiplicity of an irreducible polynomial representation of GLn(F) when restricted to Sn. We show that MPλ(ξ) embeds inside the partition algebra P|λ|(ξ). Using this embedding, we show that the multiset partition algebras are generically semisimple over F. Also, for the specialization of ξ at v in F, we prove that MPλ(v) is a cellular algebra.

The Burge correspondence yields a bijection between simple labelled graphs and semistandard Young tableaux of threshold shape. We characterize the simple graphs of hook shape by peak and valley conditions on Burge arrays. This is the first step towards an analogue of Schensted's result for the RSK insertion which states that the length of the longest increasing subword of a word is the length of the largest row of the tableau under the RSK correspondence. Furthermore, we give a crystal structure on simple graphs of hook shape. The extremal vectors in this crystal are precisely the simple graphs whose degree sequence are threshold and hook-shaped.

An irreducible character of a finite group G is called quasi p-Steinberg character for a prime p if it takes a nonzero value on every p-regular element of G. In this paper, we classify the quasi p-Steinberg characters of Symmetric (Sn) and Alternating (An) groups and their double covers. In particular, an existence of a nonlinear quasi p-Steinberg character of Sn implies n ≤ 8 and of An implies n ≤ 9.

We define two symmetric q, t-Catalan polynomials in terms of the area and depth statistic and in terms of the dinv and dinv of depth statistics. We prove symmetry using an involution on plane trees. The same involution proves symmetry of the Tutte polynomials. We also provide a combinatorial proof of a remark by Garsia et al. regarding parking functions and the number of connected graphs on a fixed number of vertices.

We construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of a general linear group to the category of representations of its Weyl group. This construction leads to a representation-theoretic proof of Littlewood’s plethystic formula for the multiplicity of an irreducible representation of the symmetric group in such a restriction. The unimodality of certain bipartite partition functions follows.

Character polynomials are used to study the restriction of a polynomial representation of a general linear group to its subgroup of permutation matrices. A simple formula is obtained for computing inner products of class functions given by character polynomials. Character polynomials for symmetric and alternating tensors are computed using generating functions with Eulerian factorizations. These are used to compute character polynomials for Weyl modules, which exhibit a duality. By taking inner products of character polynomials for Weyl modules and character polynomials for Specht modules, stable restriction coefficients are easily computed. Generating functions of dimensions of symmetric group invariants in Weyl modules are obtained. Partitions with two rows, two columns, and hook partitions whose Weyl modules have non-zero vectors invariant under the symmetric group are characterized. A reformulation of the restriction problem in terms of a restriction functor from the category of strict polynomial functors to the category of finitely generated FI-modules is obtained.

We deduce decompositions of natural representations of general linear groups and symmetric groups from combinatorial bijections involving tableaux. These include some of Howe’s dualities, Gelfand models, the Schur-Weyl decomposition of tensor space, and multiplicity-free decompositions indexed by threshold partitions.