Algebraic theory of finite fourier transforms

Beginning with the papers of G. Azumaya [2] and M. Auslander and O. Goldman [1] introducing the Brauer group B(R) of a commutative ring R, there have followed a series of extensions of the Brauer group of division algebras over a field to satisfy various needs. A. Grothendeick viewed the Brauer group of a commutative ring as the local part of a Brauer group of schemes and related the Brauer group to the second etale cohomology group of the scheme with values in the units scheaf [11]. At about the same time, C. T. C. Wall introduced in [14] a Brauer group of equivalence classes of Z/2 graded algebras over a field with multiplication induced by a twisted tensor product to study the Witt ring of quadratic forms. Wall’s construction was extended to commutative rings R by H. Bass and C. Small [13], and this group is now called the Brauer Wall group of R and denoted BW(R). A Brauer group of algebras over a field graded by an arbitrary finite abelian group was introduced by M. Knus [9], and extended to arbitrary commutative rings (with a twisted multiplication) by L. Childs, G. Garfinkel and M. Orzech [5]. The Childs-Garfinkel-Orzech construction contained the Brauer Wall group as a special case. An “equivariant Brauer group” of algebras on which a fixed group acted as a group of automorphisms was constructed by O. Frolich and C. T. C. Wall. In his thesis [10], F. W. Long introduced a Brauer group of dimodule algebras on which a grading group G acted as a group of automorphisms which included the affine versions of all the previous extensions of the Brauer group as subgroups. Long’s group is now called the Brauer Long group of R and is denoted BD(R, G). After Long introduced his group, a steady stream of papers have considered the properties and calculations of BD(R, G) and its siblings. Some of these are listed among the references at the end of this report.

The singular submodule splits off

Hopf algebras over number rings

Let U(KG) be the group of units of the group ring KG of the group G over a commutative ring K. The anti-automorphism g → g −1 of G can be extended linearly to an anti-automorphism a → a * of KG. Let S * (KG) = {x ∈ U(KG) | x * = x} be the set of all symmetric units of U(KG). We consider the following question: for which groups G and commutative rings K it is true that S * (KG) is a subgroup in U(KG). We answer this question when either a) G is torsion and K is a commutative G-favourable integral domain of characteristic p≥ 0 or b) G is non-torsion nilpotent group and KG is semiprime.

Tambara functors are an equivariant generalization of commutative rings. In previous work, Nakaoka introduced the spectrum of prime ideals of a Tambara functor and computed the spectrum of the Burnside Tambara functor, the equivariant analogue of the Zariski spectrum of the integers, over cyclic $p$-groups. Subsequently, Calle and Ginnett computed the spectrum of the Burnside Tambara functor over arbitrary finite cyclic groups using a generalization of Dress’ ghost coordinates for Burnside rings. In this paper, we compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group. Our proof uses recent advances in the commutative algebra of Tambara functors, as well as a Tambara functor analogue of ghost coordinates, which works over arbitrary finite groups and clarifies some previous computations. As examples, we explicitly compute the spectrum of the Burnside Tambara functor over all dihederal groups, the quaternion group $Q_{8}$, the alternating group $A_{4}$, and the general linear group $GL_{3}(\mathbb{F}_{2})$.

We address the study of multiparameter quantum groups (MpQGs) at roots of unity, namely quantum universal enveloping algebras U_\mathbf{q}(\mathfrak{g}) depending on a matrix of parameters \mathbf{q}=(q_{ij})_{i,j\in I} . This is performed via the construction of quantum root vectors and suitable “integral forms” of U_\mathbf{q}(\mathfrak{g}) , a restricted one—generated by quantum divided powers and quantum binomial coefficients—and an unrestricted one—where quantum root vectors are suitably renormalized. The specializations at roots of unity of either form are the “MpQGs at roots of unity” we look for. In particular, we study special subalgebras and quotients of our MpQGs at roots of unity—namely, the multiparameter version of small quantum groups—and suitable associated quantum Frobenius morphisms, that link the MpQGs at roots of 1 with MpQGs at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion, often at the core of our strategy, is that every MpQG is actually a 2 -cocycle deformation of the algebra structure of (a lift of) the “canonical” one-parameter quantum group by Jimbo–Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter \mathbf{q} , our quantum groups yield (through the choice of integral forms and their specializations) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group.

The linear complexityL K(A) of a matrixA over a fieldK is defined as the minimal number of additions, subtractions and scalar multiplications sufficient to evaluateA at a generic input vector. IfG is a finite group andK a field containing a primitive exp(G)-th root of unity,L K(G):= min{L K(A)|A a Fourier transform forKG} is called theK-linear complexity ofG. We show that every supersolvable groupG has amonomial Fourier Transform adapted to a chief series ofG. The proof is constructive and gives rise to an efficient algorithm with running timeO(|G|2log|G|). Moreover, we prove that these Fourier transforms are efficient to evaluate:L K(G)≤8.5|G|log|G| for any supersolvable groupG andL K(G)≤1.5|G|log|G| for any 2-groupG.

A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism σ of ℙ n−1, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if σ is 'generic enough', then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if σ has a representative in GL(ℂ n ) which belongs to G. As an example, the results are applied to the coordinate ring $$\mathcal{O}_q (M_2 )$$ of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of $$\mathcal{O}_q (M_2 )$$ and the symplectic leaves.

ABSTRACTIt is well known that the perfect nonlinearity of a function between finite groups and can be characterized by its graph in terms of relative difference set in the direct product (cf. [4]). Let be the infinite set of complex roots of unity. A ‐valued function on an arbitrary finite group is associated with a finite cyclic subgroup in the multiplicative group of nonzero complex numbers. For a bent function on in general, its graph is not a relative difference set in the direct product . In this paper, we investigate the necessary and sufficient conditions under which the graph of a bent function on is a relative difference set in . Cyclotomic fields and their integral bases play an important role in our discussions.

We show that a k k -stable set in a finite group can be approximated, up to given error ϵ > 0 \epsilon >0 , by left cosets of a subgroup of index ϵ - O k ( 1 ) \epsilon ^{\text {-} O_k(1)} . This improves the bound in a similar result of Terry and Wolf on stable arithmetic regularity in finite abelian groups, and leads to a quantitative account of work of the author, Pillay, and Terry on stable sets in arbitrary finite groups. We also prove an analogous result for finite stable sets of small tripling in arbitrary groups, which provides a quantitative version of recent work by Martin-Pizarro, Palacín, and Wolf. Our proofs use results on VC-dimension, and a finitization of model-theoretic techniques from stable group theory.

If R R is a commutative integral domain with quotient field K K and x 1 , … , x n {x_1}, \ldots ,{x_n} are indeterminates, then there exist θ 1 , … , θ n {\theta _1}, \ldots ,{\theta _n} in K K such that dim ⁡ R [ x 1 , … , x n ] = n + dim ⁡ R [ θ 1 , … , θ n ] \dim R[{x_1}, \ldots ,{x_n}] = n + \dim R[{\theta _1}, \ldots ,{\theta _n}] .

<strong>11.1 Generalized Hadamard Matrices</strong> The generalized Hadamard matrices were introduced by Butson in 1962. Generalized Hadamard matrices arise naturally in the study of error-correcting codes, orthogonal arrays, and affine designs (see Refs. 2-4). In general, generalized Hadamard matrices are used in digital signal/image processing in the form of the fast transform by Walsh, Fourier, and Vilenkin-Chrestenson-Kronecker systems. The survey of generalized Hadamard matrix construction can be found in Refs. 2 and 5-12. <strong>11.1.1 Introduction and statement of problems</strong> <strong>Definition 11.1.1.1:</strong> A square matrix <i>H</i>(<i>p</i>, <i>N</i>) of order <i>N</i> with elements of the <i>p</i>'th root of unity is called a generalized Hadamard matrix if <i>HH</i>^&#8727; = <i>H</i>^&#8727;<i>H</i> = <i>NI</i>_<i>N</i>, where <i>H</i>&#8727; is the conjugate-transpose matrix of <i>H</i>. <i>Remarks:</i> The generalized Hadamard matrices contain the following: • A Sylvester-Hadamard matrix if <i>p</i> = 2, <i>N</i> = 2^<i>n</i>. • A real Hadamard matrix if <i>p</i> = 2, <i>N</i> = 4<i>t</i>. • A complex Hadamard matrix if <i>p</i> = 4, <i>N</i> = 2<i>t</i>. • A Fourier matrix if <i>p</i> = <i>N</i>, <i>N</i> = <i>N</i>. Note: Vilenkin-Kronecker systems are generalized Hadamard <i>H</i>(<i>p</i>, <i>p</i>) and <i>H</i>(<i>p</i>, <i>pn</i>) matrices, respectively.

As a homomorphic image of the hyperalgebra $$U_{q,R}(m|n)$$ associated with the quantum linear supergroup $$U_{\varvec{\upsilon }}(\mathfrak {gl}_{m|n})$$, we first give a presentation for the q-Schur superalgebra $$S_{q,R}(m|n,r)$$ over a commutative ring R. We then develop a criterion for polynomial supermodules of $$U_{q,F}(m|n)$$ over a field F and use this to determine a classification of polynomial irreducible supermodules at roots of unity. This also gives classifications of irreducible $$S_{q,F}(m|n,r)$$-supermodules for all r. As an application when $$m=n\ge r$$ and motivated by the beautiful work (Brundan and Kujawa in J Algebraic Combin 18:13–39, 2003) in the classical (non-quantum) case, we provide a new proof for the Mullineux conjecture related to the irreducible modules over the Hecke algebra $$H_{q^2,F}({{\mathfrak {S}}}_r)$$; see Brundan (Proc Lond Math Soc 77:551–581, 1998) for a proof without using the super theory.

Principal Galois orders and Gelfand-Zeitlin modules

Poisson orders on large quantum groups