Suppose that G is a finite π-separable group. A classical result asserts that all irreducible characters of a Hall π-subgroup H of G extend to G if and only if H has a normal complement in G. Now, we fix a prime p and analyze when only the p-degree irreducible characters of H extend to G. 2010 Mathematics Subject Classification: 20C15 (primary), 20G40 (secondary)

§1. Representations of finite groups 1 §1.1. Definition and Examples 1 §1.2. G-modules 3 §1.3. Mashke’s Theorem 4 §1.4. Schur’s Lemma 6 §1.5. One-dimensional representations 7 §1.6. Exercises 9 §2. Irreducible representations of finite groups 10 §2.1. Characters 10 §2.2. Basic operations on representations and their characters 11 §2.3. Schur’s orthogonality relations 12 §2.4. Decomposition of t...

In this article we prove the following result: that for any two natural numbers k and `, and for all sufficiently large symmetric groups Sn, there are k disjoint sets of ` irreducible characters of Sn, such that each set consists of characters with the same degree, and distinct sets have different degrees. In particular, this resolves a conjecture most recently made by Moretó in [5]. The method...

O n January 8, 2007, just before 9 in the morning, a computer finished writing to disk about sixty gigabytes of files containing the Kazhdan-Lusztig polynomials for the split real group G of type E8. Values at 1 of these polynomials are coefficients in characters of irreducible representations of G; so all irreducible characters were written down. The biggest coefficient appearing was 11,808,80...

The main purpose of this article is to announce and provide supporting evidence for two conjectures about the characters of the Hecke algebra Hn(q) of type An_I' evaluated at elements of its Kazhdan-Lusztig basis. In addition, we prove a conjectured immanant inequality for Jacobi-Trudi matrices (definitions below) and show how our conjectures would imply stronger inequalities of a similar kind....

For a prime $p$, we determine Sylow $p$-subgroup $D$ of finite group $G$ such that the principal $p$-block $B$ has four irreducible ordinary characters. It been determined already for cases where number is up to three by work R. Brauer, J. Brandt, and V.A. Belonogov thirty years ago. Our proof relies on classification simple groups.

Suppose that $\chi_\lambda$ and $\chi_\mu$ are distinct irreducible characters of the symmetric group $S_n$. We give an algorithm that, in time polynomial $n$, constructs $\pi\in S_n$ such $\chi_\lambda(\pi)$ is provably different from $\chi_\mu(\pi)$. In fact, we show a little more. $f = \chi_\lambda$ for some character $S_n$, but do not know $\lambda$, given only oracle access to $f$. determi...