Characters and quantum reduction for orthosymplectic Lie superalgebras

In this study, we study principal admissible representations for the affine Lie superalgebra [Formula: see text]. Using the character formula of irreducible admissible representations of [Formula: see text], we calculate a character formula of [Formula: see text]-modules which are obtained from the quantized Drinfeld–Sokolov reduction and principal admissible representations. As a by-product, we obtain the minimal series modules of the Neveu–Schwarz algebra through the [Formula: see text]-modules arising from the principal admissible modules over [Formula: see text].

A note on prime number races and zero free regions for L functions

Let [Formula: see text] be a real and non-principal Dirichlet character, [Formula: see text] its Dirichlet [Formula: see text]-function and let [Formula: see text] be a generic prime number. We prove the following result: If for some [Formula: see text] the partial sums [Formula: see text] change sign only for a finite number of integers [Formula: see text], then there exists [Formula: see text] such that [Formula: see text] has no zeros in the half plane [Formula: see text].

LECTURES ON THE GEOMETRY AND MODULAR REPRESENTATION THEORY OF ALGEBRAIC GROUPS

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

Finite groups with at most six vanishing conjugacy classes

Let [Formula: see text] be a finite group. We say that a conjugacy class of [Formula: see text] in [Formula: see text] is vanishing if there exists some irreducible character [Formula: see text] of [Formula: see text] such that [Formula: see text]. In this paper, we show that finite groups with at most six vanishing conjugacy classes are solvable or almost simple groups.

On Petersson’s partition limit formula

For each prime [Formula: see text] consider the Legendre character [Formula: see text]. Let [Formula: see text] be the number of partitions of [Formula: see text] into parts [Formula: see text] such that [Formula: see text]. Petersson proved a beautiful limit formula for the ratio of [Formula: see text] to [Formula: see text] as [Formula: see text] expressed in terms of important invariants of the real quadratic field [Formula: see text]. But his proof is not illuminating and Grosswald conjectured a more natural proof using a Tauberian converse of the Stolz–Cesàro theorem. In this paper, we suggest an approach to address Grosswald’s conjecture. We discuss a monotonicity conjecture which looks quite natural in the context of the monotonicity theorems of Bateman–Erdős.

Nonsolvable groups with four character codegrees

Let [Formula: see text] be a finite group and the codegree of an irreducible character [Formula: see text] is the number [Formula: see text]. In this paper, we consider nonsolvable groups with few character codegrees and prove that if a nonsolvable group [Formula: see text] has exactly four character codegrees, then [Formula: see text], where [Formula: see text].