Invariants of finite Hopf algebras

This paper extends classical results in the invariant theory of finite groups and finite group schemes to the actions of finite Hopf algebras on commutative rings. Topics considered include integrality over the invariant rings, properties of the canonical map between the prime spectra, orbital and stabilizer algebras, projectivity over the invariant rings, and descent of Cohen-Macaulayness.     

A new link between the descent algebra of type B,domino tableaux and Chow’s quasisymmetric functions

Introduced by Solomon in his 1976 paper, the descent algebra of a finite Coxeter group received significant attention over the past decades. As proved by Gessel, in the case of the symmetric group its structure constants give the comultiplication table for the fundamental basis of quasisymmetric functions. We show that this latter property actually implies several well known relations linked to the Robinson–Schensted–Knuth correspondence and some of its generalisations. This provides a new link between these results and the theory of quasisymmetric functions and allows to derive more advanced formulae involving Kronecker coefficients. Using the theory of type B quasisymmetric functions introduced by Chow, we extend this connection to the hyperoctahedral group and derive new formulae relating the structure constants of the descent algebra of type B, the numbers of domino tableaux of given descent set and the Kronecker coefficients of the hyperoctahedral group.     

Shintani lifts and fractional derivatives for harmonic weak Maass forms

In this paper, we construct Shintani lifts from integral weight weakly holomorphic modular forms to half-integral weight weakly holomorphic modular forms. Although defined by different methods, these coincide with the classical Shintani lifts when restricted to the space of cusp forms. As a side effect, this gives the coefficients of the classical Shintani lifts as new cycle integrals. This yields new formulas for the L-values of Hecke eigenforms. When restricted to the space of weakly holomorphic modular forms orthogonal to cusp forms, the Shintani lifts introduce a definition of weakly holomorphic Hecke eigenforms. Along the way, auxiliary lifts are constructed from the space of harmonic weak Maass forms which yield a “fractional derivative” from the space of half-integral weight harmonic weak Maass forms to half-integral weight weakly holomorphic modular forms. This fractional derivative complements the usual ξ-operator introduced by Bruinier and Funke.     

The six operations for sheaves on Artin stacks I: Finite coefficients

In this paper we develop a theory of Grothendieck’s six operations of lisse-étale constructible sheaves on Artin stacks locally of finite type over certain excellent schemes of finite Krull dimension. We also give generalizations of the classical base change theorems and Kunneth formula to stacks, and prove new results about cohomological descent for unbounded complexes.     

The number of flags in finite vector spaces: asymptotic normality and Mahonian statistics

We study the generalized Galois numbers which count flags of length r in N-dimensional vector spaces over finite fields. We prove that the coefficients of those polynomials are asymptotically Gaussian normally distributed as N becomes large. Furthermore, we interpret the generalized Galois numbers as weighted inversion statistics on the descent classes of the symmetric group on N elements and identify their asymptotic limit as the Mahonian inversion statistic when r approaches ∞. Finally, we apply our statements to derive further statistical aspects of generalized Rogers–Szeg? polynomials, reinterpret the asymptotic behavior of linear q-ary codes and characters of the symmetric group acting on subspaces over finite fields, and discuss implications for affine Demazure modules and joint probability generating functions of descent-inversion statistics.     

Shintani lifting and real-valued characters

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over ${\mathbb{F}_q}$ , and ψ is an irreducible character of G( ${\mathbb{F}_{q^m}}$ ) which is the lift of an irreducible character χ of G( ${\mathbb{F}_q}$ ), we prove ψ is real-valued if and only if χ is real-valued. In the case m = 2, we show that if χ is invariant under the twisting operator of G( ${\mathbb{F}_{q^2}}$ ), and is a real-valued irreducible character in the image of lifting from G( ${\mathbb{F}_q}$ ), then χ must be an orthogonal character. We also study properties of the Frobenius–Schur indicator under Shintani lifting of regular, semisimple, and irreducible Deligne–Lusztig characters of finite reductive groups.     

On the unipotent characters of the Ree groups of type <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript>

This note is concerned with the unipotent characters of the Ree groups of type G ₂. We determine the roots of unity associated by Lusztig and Digne-Michel to each unipotent character for and prove that the Fourier matrix of defined by Geck and Malle satisfies a conjecture of Digne-Michel. Our main tool is the Shintani descent of Ree groups of type G ₂.     

Finite descent obstruction for curves over function fields

We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of integral points on integral models of twists of modular curves over function fields.     

Spectral properties of descent algebra elements

In this article, we determine the eigenvalues and their corresponding multiplicities of the action on the group algebra of a finite Coxeter group of an element of its descent algebra. Meanwhile, we identify a slight error in the paper of Bergeron, Bergeron, Howlett, and Taylor in a formula for certain structure coefficients in the descent algebra. We provide the correct formula, and give an example which explicitly shows the difference between both formulas.     

Quadratic base change of θ10

In case ofGL _n overp-adic fields, it is known that Shintani base change is well behaved. However, things are not so simple for general reductive groups. In the first part of this paper, we present a counterexample to the existence of quadratic base change descent for some Galois invariant representations. These are representations of type θ₁₀. In the second part, we compute the localL-factor of θ₁₀. Unlike many other supercuspidal representations, we find that theL-factor of θ₁₀ has two poles. Finally, we discuss these two results in relation to the local Langlands correspondence. The authors are supported in part by NSF grants.     

Finite preorders and topological descent II: étale descent

It is known that every effective (global-) descent morphism of topological spaces is an effective étale-descent morphism. On the other hand, in the predecessor of this paper we gave examples of:

•

a descent morphism that is not an effective étale-descent morphism;

•

an effective étale-descent morphism that is not a descent morphism.

Both of the examples in fact involved only finite topological spaces, i.e. just finite preorders, and now we characterize the effective étale-descent morphisms of preorders/finite topological spaces completely.     

A quiver presentation for Solomon's descent algebra

The descent algebra Σ(W) is a subalgebra of the group algebra QW of a finite Coxeter group W, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of W. Thus Σ(W) is a basic algebra, and as such it has a presentation as a quiver with relations. Here we construct Σ(W) as a quotient of a subalgebra of the path algebra of the Hasse diagram of the Boolean lattice of all subsets of S, the set of simple reflections in W. From this construction we obtain some general information about the quiver of Σ(W) and an algorithm for the construction of a quiver presentation for the descent algebra Σ(W) of any given finite Coxeter group W.     

The p-Modular Descent Algebra of the Symmetric Group

The descent algebra of the symmetric group, over a field ofnon-zero characteristic p, is studied. A homomorphism into thealgebra of generalised p-modular characters of the symmetricgroup is defined. This is then used to determine the radical,and its nilpotency index. It also allows the irreducible representationsof the descent algebra to be described. 1991 Mathematics SubjectClassification 20F32.     

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields.

In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

     

A Decomposition of the Descent Algebra of a Finite Coxeter Group

The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type A _n and B _n. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14], is a subalgebra of the group algebra of W. It is closely related to the subring of the Burnside ring B(W) spanned by the permutation representations W/W _J, where the W _J are the parabolic subgroups of W. Specifically, our purpose is to lift a basis of primitive idempotents of the parabolic Burnside algebra to a basis of idempotents of the descent algebra.     

A successive descent algorithm over a system of local minima

A successive descent algorithm over a system of local minima has been developed to find the global minimum of a function of many variables defined on a simply connected compact set. If the number of local minima is finite and a bound on the global minimum is given, the algorithm finds the global minimum in finitely many steps. Test examples are presented. Translated from Prikladnaya Matematika i Informatika, No. 30, pp. 46–54, 2008.     

Projective normality of G.I.T. quotient varieties modulo finite groups

We prove that for any finite dimensional representation V of a finite group G of order n the quotient variety G??(V) is projectively normal with respect to descent of 𝒪(1)^?l where l = lcm{1,2,3,4,…,n}. We also prove that for the tautological representation V of the alternating group A_n the projective variety A_n??(V) is projectively normal with respect to the descent of the above line bundle.     

Skew exactness perturbation

We offer a perturbation theory for finite ascent and descent properties of bounded operators.