A note on the Hambleton-Taylor-Williams conjecture

I. Hambleton, L. Taylor and B. Williams conjectured a general formula in the spirit of H. Lenstra for the decomposition of $G_n(RG)$ for any finite group G and noetherian ring R. The conjectured decomposition was shown to hold for some large classes of finite groups. D. Webb and D. Yao discovered that the conjecture failed for the symmetric group $S_5$, but remarked that it still might be reasonable to expect the HTW-decomposition for solvable groups. In this paper we show that the solvable group $\mathrm{SL}(2,{\mathbb{F}}_3)$ is also a counterexample to the conjectured HTW-decomposition. Nevertheless, we prove that for any finite group G the rank of $G_1(\mathbb{Z}G)$ does not exceed the rank of the expression in the HTW-decomposition. We also show that the HTW-decomposition predicts correct torsion for $G_1(\mathbb{Z}G)$ for any finite group G. Furthermore, we prove that for any degree other than $n=1$ the conjecture gives a correct prediction for the rank of $G_n(\mathbb{Z}G)$.     

The Whitehead group of (almost) extra-special p-groups with p odd

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup $C{l}_1(\mathbb{Z}P)$ of $S{K}_1(\mathbb{Z}P)$, in terms of a genetic basis of P. We also introduce a deflation map $C{l}_1(\mathbb{Z}P)\to C{l}_1(\mathbb{Z}(P/N))$, for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of $S{K}_1(\mathbb{Z}P)$, when P is an elementary abelian p-group.     

Graded-division algebras and Galois extensions

Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras.On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field $\mathbb{F}$ is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of $\mathbb{F}$ and the isomorphism class of a G-Galois extension of $\mathbb{F}$.This connection is used to classify the simple G-Galois extensions of $\mathbb{F}$ in terms of a Galois field extension $\mathbb{L}/\mathbb{F}$ with Galois group isomorphic to a quotient $G/K$ and an element in the quotient ${\mathrm{\text{Z}}}^2(K,{\mathbb{L}}^{\times })/{\mathrm{\text{B}}}^2(K,{\mathbb{F}}^{\times })$ subject to certain conditions. Non-simple G-Galois extensions are induced from simple T-Galois extensions for a subgroup T of G. We also classify finite-dimensional G-graded-division algebras and, as an application, finite G-graded-division rings.     

Integral representations of martingales for progressive enlargements of filtrations

We work in the setting of the progressive enlargement $\mathbb{G}$ of a reference filtration $\mathbb{F}$ through the observation of a random time $\tau$. We study an integral representation property for some classes of $\mathbb{G}$-martingales stopped at $\tau$. In the first part, we focus on the case where $\mathbb{F}$ is a Poisson filtration and we establish a predictable representation property with respect to three $\mathbb{G}$-martingales. In the second part, we relax the assumption that $\mathbb{F}$ is a Poisson filtration and we assume that $\tau$ is an $\mathbb{F}$-pseudo-stopping time. We establish integral representations with respect to some $\mathbb{G}$-martingales built from $\mathbb{F}$-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two $\mathbb{G}$-martingales.     

When is scalar multiplication decidable?

Let K be a subfield of $\mathbb{R}$. The theory of $\mathbb{R}$ viewed as an ordered K-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if K is a real quadratic field.     

Universal deformation rings and self-injective Nakayama algebras

Let k be a field and let Λ be an indecomposable finite dimensional k-algebra such that there is a stable equivalence of Morita type between Λ and a self-injective split basic Nakayama algebra over k. We show that every indecomposable finitely generated Λ-module V has a universal deformation ring $R(\mathrm{\Lambda },V)$ and we describe $R(\mathrm{\Lambda },V)$ explicitly as a quotient ring of a power series ring over k in finitely many variables. This result applies in particular to Brauer tree algebras, and hence to p-modular blocks of finite groups with cyclic defect groups.     

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finite residue field $\mathsf{k}$ of odd characteristic, and let G be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $?\in \mathbb{N}$ let $G^{?}$ denote the ?-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{?+1}$ for some ?, and if its restriction to $G^{?}/G^{?+1}?\mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}({\mathsf{k}}^{\mathrm{alg}})$-stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.     

The extension theorem for the Lee and Euclidean weights over Z/pkZ

The Lee and Euclidean weights have the extension property over the local rings $\mathbb{Z}/p^k\mathbb{Z}$, p prime. The non-vanishing of certain Fourier coefficients is established by expressing the coefficients in terms of generalized Bernoulli numbers and making use of knowledge of the locations of zeros of Dirichlet L-functions.     

Even degree characters in principal blocks

We characterise finite groups such that for an odd prime p all the irreducible characters in its principal p-block have odd degree. We show that this situation does not occur in non-abelian simple groups of order divisible by p unless $p=7$ and the group is $M_{22}$. As a consequence we deduce that if $p\ne 7$ or if $M_{22}$ is not a composition factor of a group G, then the condition above is equivalent to $G/{\mathbf{O}}_{p^{\prime }}(G)$ having odd order.     

Gorenstein projective bimodules via monomorphism categories and filtration categories

We generalize the monomorphism category from quiver (with monomial relations) to arbitrary finite dimensional algebras by a homological definition. Given two finite dimension algebras A and B, we use the special monomorphism category $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ to describe some Gorenstein projective bimodules over the tensor product of A and B. If one of the two algebras is Gorenstein, we give a sufficient and necessary condition for $\mathsf{Mon}(B,A\text{-}\mathsf{Gproj})$ being the category of all Gorenstein projective bimodules. In addition, if both A and B are Gorenstein, we can describe the category of all Gorenstein projective bimodules via filtration categories. Similarly, in this case, we get the same result for infinitely generated Gorenstein projective bimodules.     

On the weak representation property in progressively enlarged filtrations with an application in exponential utility maximization

In this paper we show that the weak representation property of a semimartingale $X$ with respect to a filtration $\mathbb{F}$ is preserved in the progressive enlargement $\mathbb{G}$ by a random time $\tau$ avoiding $\mathbb{F}$-stopping times and such that $\mathbb{F}$ is immersed in $\mathbb{G}$. As an application of this, we can solve an exponential utility maximization problem in the enlarged filtration $\mathbb{G}$ following the dynamical approach, based on suitable BSDEs, both over the fixed-time horizon $[0,T]$, $T>0$, and over the random-time horizon $[0,T\wedge \tau ]$.     

Bounds on multiplicities of spherical spaces over finite fields

Let G be a reductive group scheme of type A acting on a spherical scheme X. We prove that there exists a number C such that the multiplicity $\dim \mathrm{Hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by C, for any finite field F and any irreducible representation ρ of $G(F)$. We give an explicit bound for C. We conjecture that this result is true for any reductive group scheme and when F ranges (in addition) over all local fields of characteristic 0.Different aspects of this conjecture were studied in [3], [11], [6], [7].     

C-system of a module over a Jf-relative monad

Let $\mathbb{F}$ be the category with the set of objects $\mathbb{N}$ and morphisms given by the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:\mathbb{F}\to Sets(U)$ be the obvious functor from this category to the category of sets in a given Grothendieck universe U. In this paper we construct, for any Jf-relative monad RR and any left RR-module LM, a C-system $C(\mathbf{RR},\mathbf{LM})$ and explicitly compute the action of the four B-system operations on its B-sets.In the introduction we explain in detail the relevance of this result to the construction of the term C-systems of type theories.