Combinatorics
The representation theory of the symmetric group is a meeting point of representation theory and combinatorics. The combinatorics encountered when dealing with the representation theory of the symmetric groups is one of my interests. The two directions that my research here has taken are understanding the degrees of irreducible characters, and understanding the number of blocks of defect zero.
The character degrees of the symmetric groups are combinatorially described by a famous formula of Frame, Robinson, and Thrall [3], building on the pioneering work of Frobenius. One question which is of interest is to ask how many different irreducible characters of the symmetric group Sn have the same degree. The largest multiplicity of the character degrees will be denoted by m(n).
As we mentioned, the character degrees are determined combinatorially, by associating to each partition of n, a multiset of integers called the hook numbers. Two characters have the same degree if and only if the two corresponding multisets of hook numbers have the same product. Deciding whether two multisets of integers have the same product is a difficult number-theoretic problem, and I decided that it might be easier to simply ask whether the two multisets were equal. Denote by M(n) the largest multiplicity of a multiset of hook numbers, as one ranges of the partitions of n. Clearly, m(n) is at least as large as M(n).
Theorem (Craven [1]). The function M(n) tends to infinity as n tends to infinity. Consequently, the function m(n) also tends to infinity as n does.
One may imagine a corresponding function m(G), where G is a finite group. (Thus m(G) is the maximal multiplicity of the degrees of the irreducible characters.) We have the following corollary, the culmination of work by several mathematicians.
Corollary (Craven [1], Jaikin-Zapirain [4], Moretó [5]). The order of a finite group G is bounded in terms of m(G).
One of the directions of research that I am pursuing at the moment is attempting to determine explicit bounds in this corollary. As Moretó has mentioned in [5], this can be done. However, I am interested in what the actual growth of m(G) is for general G, as well as for specific families of groups.
The multiset of hook numbers also determines whether the ordinary character lies in a block of defect zero or not. Let p be a prime; if none of the hook numbers of a partition is divisible by p, we say that the partition is its own p-core. This notion may be extended to non-primes, in which case we often used the letter 't', instead. If a partition is its own t-core, then the corresponding ordinary character lies in a t-block of defect zero. Thus knowing which partitions are t-cores is an interesting problem.
Denote by ct(n) the number of partitions of n that are t-cores. In [6], Stanton conjectured that the equation
ct(n)≤ct+1(n)
holds for all t≥4. For one particular value of n, namely t=n-1, this does not hold, but it appears to hold for al other t and n.
Theorem (Craven [2]). Suppose that t>n/2. Then
ct(n)<ct+1(n)
It appears to be difficult to extend this result for smaller values of t, although with better bounds on the partition function, this might be possible using the methods in this paper.
References
[1] David A. Craven, Symmetric Group Character Degrees and Hook Numbers, Proc. Lond. Math. Soc. (3) 96 (2008), no. 1, 26–50. MathSciNet, PDF
[2] David A. Craven, The Number of t-Cores of Size n, preprint. PDF
[3] J. Sutherland Frame, Gilbert de B. Robinson, and Robert M. Thrall, The Hook Graphs of the Symmetric Groups, Canadian J. Math. 6 (1954), 316–324. MathSciNet
[4] Andrei Jaikin-Zapirain, On the Number of Conjugacy Classes of Finite p-Groups, J. London Math. Soc. (2) 68 (2003), no. 3, 699–711. MathSciNet
[5] Alexander Moretó, Complex Group Algebras of Finite Groups: Brauer's Problem 1, Adv. Math. 208 (2007), no. 1, 236–248. MathSciNet
[6] Dennis Stanton, Open Positivity Conjectures for Integer Partitions, Trends Math. 2 (1999), 19–25 (electronic). DVI from Stanton's website