A New Proof of the Mullineux Conjecture

Let S _d denote the symmetric group on d letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible S _d-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchev's approach. Applying the representation theory of the supergroup GL(m | n) and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineux's conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of GL(m | n) of degree d for arbitrary m, n, and d.     

Nilpotency in type <Emphasis Type="Italic">A</Emphasis> cyclotomic quotients

We prove a conjecture made by Brundan and Kleshchev on the nilpotency degree of cyclotomic quotients of rings that categorify one-half of quantum sl(k).     

Irreducible representations of the alternating group in odd characteristic

We use the recently-proved conjecture of Mullineux to determine which modular irreducible representations of the symmetric group split on restriction to , and which remain irreducible (everything taking place over a splitting field for of characteristic ). An indexing of the absolutely irreducible representations of is thus obtained. A modular analogue of the Frobenius symbol for a partition is introduced, which makes the Mullineux map somewhat more intuitive.

     

On the Mullineux involution for Ariki–Koike algebras

This note is concerned with a natural generalization of the Mullineux involution for Ariki–Koike algebras. By using a result of Fayers together with previous results by the authors, we give an efficient algorithm for computing this generalized Mullineux involution. Our algorithm notably does not involve the determination of paths in affine crystals.     

All group‐based latin squares possess near transversals

In a latin square of order n, a near transversal is a collection of n ?1 cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.     

On Cartan matrices and lower defect groups for covering groups of symmetric groups

We determine the elementary divisors of the Cartan matrices of spin -blocks of the covering groups of the symmetric groups when is an odd prime. As a consequence, we also compute the determinants of these Cartan matrices, and in particular we confirm a conjecture by Brundan and Kleshchev that these determinants depend only on the weight but not on the sign of the block.

     

Distinct r-tuples in integer partitions

The Ramanujan Journal - Let $$p&gt;3$$ be a prime and let a be a positive integer. We show that if or $$a&gt;1$$ , then with $$(-)$$ the Jacobi symbol, which confirms a conjecture of Z.-W....     

Cellular-indecomposable subnormal operators II

This work continues that begun in [9]. Our investigation has led us to the following conjecture: a cyclic subnormal operator is cellular-indecomposable if and only if it is quasi-similar to an analytic Toeplitz operator whose symbol is a weak-star generator of H. In this paper some particular cases of the conjecture are verified.This work was supported in part by a grant from the National Science Foundation.     

On an analogue of a conjecture of Gross

The paper is concerned with a problem in the theory of congruence function fields which is analogous to a conjecture of Gross in Iwasawa Theory. Z_p-extensions K_∞/K₀ of congruence function fields K₀ of characteristic p≠2 involving no new constants are considered such that the set S of ramified primes is finite and these primes are fully ramified. Is the set of S-classes invariant under Gal(K_∞/K₀) finite ? Gross' conjecture asserts that a similar question has an affirmative answer for the class of cyclotomic Z_p- extensions of CM-type if S is the set of p-primes and the classes considered are minus S-classes. Using a formula of Witt for the norm residue symbol in cyclic p-extensions of local fields of characteristic p, a necessary and sufficient condition for the validity of the analogue of Gross' conjecture is given for a class of extensions K_∞/K₀. It is shown by examples that the analogue of Gross' conjecture is not always true.     

Eigenvectors of Hessenberg Toeplitz matrices and a problem by Dai,Geary, and Kadanoff

The paper is devoted to the eigenvectors of Hessenberg Toeplitz matrices whose symbol has a power singularity. We describe the structure of the eigenvectors and prove an asymptotic formula which can be used to compute individual eigenvectors effectively. The symbols of our matrices are special Fisher–Hartwig symbols, and the theorem of this paper confirms and makes more precise a conjecture by Dai, Geary, and Kadanoff of 2009 in a particular case.     

p-adic periods and modular symbols of elliptic curves of prime conductor

Under certain assumptions, we prove a conjecture of Mazur and Tate describing a relation between the modular symbol attached to an elliptic curve with split multiplicative reduction atp, and itsp-adic period. We generalize this relation to modular forms of weight 2 with coefficients not necessarily in.Oblatum 24-XI-1993 & 8-VI-1994     

Latin Squares and the Hall-Paige Conjecture

The Hall–Paige conjecture deals with conditions underwhich a finite group G will possess a complete mapping, or equivalentlya Latin square based on the Cayley table of G will possess atransversal. Two necessary conditions are known to be: (i) thatthe Sylow 2-subgroups of G are trivial or non-cyclic, and (ii)that there is some ordering of the elements of G which yieldsa trivial product. These two conditions are known to be equivalent,but the first direct, elementary proof that (i) implies (ii)is given here. It is also shown that the Hall–Paige conjecture impliesthe existence of a duplex in every group table, thereby provinga special case of Rodney's conjecture that every Latin squarecontains a duplex. A duplex is a ‘double transversal’,that is, a set of 2n entries in a Latin square of order n suchthat each row, column and symbol is represented exactly twice.2000 Mathematics Subject Classification 05B15, 20D60.     

Derandomizing Restricted Isometries via the Legendre Symbol

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special—our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction.     

The number of simple modules of the Hecke algebras of type G(r,1,n)

This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r,1,n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of “Kleshchev multipartitions”. Received July 24, 1998; in final form February 8, 1999     

Representations of quiver Hecke algebras via Lyndon bases

A new class of algebras has been introduced by Khovanov and Lauda and independently by Rouquier. These algebras categorify one-half of the Quantum group associated to arbitrary Cartan data. In this paper, we use the combinatorics of Lyndon words to construct the irreducible representations of those algebras associated to Cartan data of finite type. This completes the classification of simple modules for the quiver Hecke algebra initiated by Kleshchev and Ram.     

Quasi-classical asymptotics for pseudodifferential operators with discontinuous symbols: Widom’s conjecture

In 1982 H. Widom conjectured a multi-dimensional generalization of a well-known two-term quasi-classical asymptotic formula for the trace of the function f(A) of a Wiener—Hopf-type operator A in dimension 1 for a pseudodifferential operator A with symbol a(x, ξ) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs in a hyperplane.     

Weak closure and Oliver’s p-group conjecture

To date almost all verifications of Oliver’s p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3⁴⁹ which refutes the weakly closed conjecture but satisfies Oliver’s conjecture.     

Bethe-Sommerfeld conjecture for periodic operators with strong perturbations

We consider a periodic self-adjoint pseudo-differential operator H=(?Δ) ^m +B, m>0, in ? ^d which satisfies the following conditions: (i) the symbol of B is smooth in x, and (ii) the perturbation B has order less than 2m. Under these assumptions, we prove that the spectrum of H contains a half-line. This, in particular implies the Bethe-Sommerfeld conjecture for the Schrödinger operator with a periodic magnetic potential in all dimensions.     

Equivariant local Epsilon Constants and Etale Cohomology

A conjecture is formulated which relates the equivariant localepsilon constant of a Galois extension of p-adic fields to anatural algebraic invariant coming from étale cohomology.Some evidence for the conjecture is provided and its relationto a conjecture for the equivariant global epsilon constantof an extension of number fields formulated by Bley and Burnsis established.