Extraspecial towers and weil representations

Weil's representation is a basic object in representation theory which plays a crucial role in many places: construction of unitary irreducible representations in the frame of the orbit method, Howe correspondence, Theta series, … The decomposition in irreducibles of the restriction of Weil's representation to maximal compact subgroups or anisotropic tori of the metaplectic group is thus an important information in representation theory. Except for $SL(2)$, this was not known in the p-adic case. In this article, we prove that the restriction of the Weil representation over a p-adic field, $p\neq2$, to maximal compact subgroups is multiplicity free and give an explicit description of the irreducibles occurring. In another paper, using our results, we describe the decomposition of the restriction of the Weil representation to maximal elliptic tori.

Introduction. In the investigation of a non-degenerate quadratic form f (x) with coefficients in a local or global field K, one associates two groups with f (x), the orthogonal and metaplectic groups. If K is, e.g., the field R of real numbers, the second group is the unitary group generated by Ut = exp(it -f(x)) and its conjugate V, under the Fourier transformation for all t in R; and it is locally isomorphic to SL2(R). By using a similar definition one can associate with any polynomial f (x) of higher degree a group. Such groups have appeared in connection with the Siegel-Weil formulas for the norm forms of certain simple Jordan algebras. On the other hand Kubota [4], [5] and Yamazaki [8] have associated for K = C and R metaplectic groups with where n need not be 2, in fact n 2 and n = 2k 2, respectively. In both cases they have used certain Bessel transformations instead of the Fourier transformation to get the metaplectic groups. In a recent paper [3] we have shown that the Lie algebra of a hypermetaplectic group over R is infinite dimensional (at least if the Fourier transformation is defined relative to the bilinear form associated with a definite quadratic form). In this paper we have restricted ourselves to the one-variable case and tried to obtain all metaplectic groups over K = R and C. More precisely we have started from an arbitrary polynomial f (x) of one variable x with coefficients in K = R or C, put U, = exp (it .f (x)) or exp(2iRe(tf(x)) and determined all cases where U, and another oneparameter subgroup V, of the unitary group of a certain L2-space generate a Lie group G locally isomorphic to SL2(K). In doing so we have assumed that the tangent vector or vectors of V, at t = 0 is a linear differential operator with polynomial coefficients, a condition satisfied in the known cases. The result is that f (x) simply becomes x, x2 under a suitable normalization and, accordingly, the differential operator becomes i-times

The study of groups has been of interest to mathematicians since the 19th century. Although much is known about the structure of groups, many group theoretic problems remain unsolved. Representation theory allows us to employ linear algebra to solve such problems. The representation theory of linear groups over finite fields has been a particularly interesting topic. Studying these representations is of interest to mathematicians and other scientists as it relates to physics and modern number theory. In the 1960s Andre Weil introduced a method for finding a special unitary representation for symplectic groups over locally compact fields. This unitary representation is now referred to as the Weil representation. In 2010 Luis Gutierrez, Jose Pantoja and Jorge Soto-Andrade were able to generalize Weil’s method to a larger class of linear groups namely the ∗-analogue of Sl2. Originally, Weil constructed this unitary representation, decomposed it into irreducibles and, in this way, produced the irreducible complex representations of Sp(2n, k). Later, Shalika went in the other direction, first finding the irreducible representations and then computing their multiplicities in the Weil representation. We intend to follow Shalika’s method. In this thesis we look to explore the representation theory of Sl∗(2,A) where A is the direct sum of the upper and lower n × n block matrices in M(2n, k), k a finite field. We use Wigner and Mackey’s Method of Little Groups to construct these representations.

The notion of globally irreducible representations of finite groups was introduced by B.H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell–Weil lattices of elliptic curves. It has been observed by R. Gow and Gross that irreducible Weil representations of certain finite classical groups lead to globally irreducible representations. In this paper we classify all globally irreducible representations coming from Weil representations of finite classical groups.

This book grew out of seminar held at the University of Paris 7 during the academic year 1985-86. The aim of the seminar was to give an exposition of the theory of the Metaplectic Representation (or Weil Representation) over a p-adic field. The book begins with the algebraic theory of symplectic and unitary spaces and a general presentation of metaplectic representations. It continues with exposes on the recent work of Kudla (Howe Conjecture and induction) and of Howe (proof of the conjecture in the unramified case, representations of low rank). These lecture notes contain several original results. The book assumes some background in geometry and arithmetic (symplectic forms, quadratic forms, reductive groups, etc.), and with the theory of reductive groups over a p-adic field. It is written for researchers in p-adic reductive groups, including number theorists with an interest in the role played by the Weil Representation and -series in the theory of automorphic forms.

We study the mod ℓ \ell Weil representation of a finite unitary group and related Deligne–Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod ℓ \ell Howe correspondence for ( S p 2 n , O 2 − ) (\mathrm {Sp}_{2n},\mathrm {O}_2^-) including the case where p = 2 p=2 .

In this paper we describe connections among extraspecial 2-groups, unitary representa-tions of the braid group and multi-qubit braiding quantum gates. We first construct newrepresentations of extraspecial 2-groups. Extending the latter by the symmetric group,we construct new unitary braid representations, which are solutions to generalized Yang-Baxter equations and use them to realize new braiding quantum gates. These gates gen-erate the GHZ (Greenberger-Horne-Zeilinger) states, for an arbitrary (particularly anodd) number of qubits, from the product basis. We also discuss the Yang-Baxterizationof the new braid group representations, which describes unitary evolution of the GHZstates. Our study suggests that through their connection with braiding gates, extraspe-cial 2-groups and the GHZ states may play an important role in quantum error correctionand topological quantum computing.

A theory of frames that extend Gabor analysis by including chirping is discussed. The chirping parameterin these 'time-frequency localization frames' depends on time and/or frequency shift parameters that can beadapted to analyze and detect chirps in noisy signals. Radar/sonar applications are outlined. The frame theoryis motivated by a generalized notion of square-integrable group representation developed by Au, Antoine, andGazeau, together with ideas in Baraniuk's thesis on a metaplectic extension of Cohen's class.Keywords: chirps, frames, metaplectic group, time-frequency localization 1 INTRODUCTION Versatile time-frequency/time-scale algorithms like best basis or matching pursuits are excellent for denoising if one is content to let signal and noise be defined by the algorithm. However, when searching for specific typesof signals embedded in noise at high levels, one must possess a concrete a-priori idea of the signal one seeks. For such detection problems the philosophy of using a dictionary of time frequency localized patterns still makes

Let p p be a prime. We denote by S n S_n the symmetric group of degree n n , by A n A_n the alternating group of degree n n and by F p {\mathbb F}_p the field with p p elements. An important concept of modular representation theory of a finite group G G is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring F q G {\mathbb F}_q G , where q q is a prime power. Here, we describe a new method to compute the primitive central idempotents of F q G {\mathbb F}_q G for arbitrary prime powers q q and arbitrary finite groups G G . For the group rings F p S n {\mathbb F}_p S_n of the symmetric group, we show how to derive the primitive central idempotents of F p S n − p {\mathbb F}_p S_{n-p} from the idempotents of F p S n {\mathbb F}_p S_n . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of F p S n {\mathbb F}_p S_n which contains the primitive central idempotents. The described results are most efficient for p = 2 p = 2 . In an appendix we display all primitive central idempotents of F 2 S n {\mathbb F}_2 S_n and F 4 A n {\mathbb F}_4 A_n for n ≤ 50 n \le 50 which we computed by this method.

The symmetric group given by a Gröbner basis

Let be the cyclic group of order m and N be the direct product of n copies of . Let Sn be the symmetric group of degree n. The wreath product of with Sn is a split extension of N by Sn, called the generalized symmetric group, here denoted by B(m, n). In his Ph.D. thesis Almestady presented a combinatorial method for constructing the Fischer-Clifford matrices of B(m, n). However as a few examples for small values of m and n show, the manual calculation of these matrices presents formidable problems and hence a computerized approach to this combinatorial method is necessary. In a previous paper the current authors have given a computer programme that computes matrices which are row equivalent to the Fischer-Clifford matrices of B(2, n). Here that programme is generalized to B(m, n), where m is any positive integer. It is anticipated that with some improvements, a number of the programmes given here can be incorporated into GAP. Indeed with further development work these programmes should lead to an alternative method for computing the character table of B(m, n) in GAP.

We construct two kinds of presentations for the alternating and symmetric groups of degree n n : the first are on two generators in which the number of relations is O ( log ⁡ n ) O(\log n) and the presentation length is O ( log 2 ⁡ n ) O(\log ^2 n) ; the second have a bounded number of generators and relations and length O ( log ⁡ n ) O(\log n) .

Symmetric 1-factorizations of the complete graph

The uniqueness of a certain graph

Let\n$G$ be a subgroup of\n$S_{6}$ , the symmetric group of degree 6. For any field\n$k$ ,\n$G$ acts naturally on the rational function field\n$k(x_{1},\\ldots,x_{6})$ via\n$k$ -automorphisms defined by\n$\\sigma\\cdot x_{i}=x_{\\sigma(i)}$ for any\n$\\sigma\\in G$ and any\n$1\\le i\\le 6$ . We prove the following theorem. The fixed field\n$k(x_{1},\\ldots,x_{6})^{G}$ is rational (i.e., purely transcendental) over\n$k$ , except possibly when\n$G$ is isomorphic to\n$\\operatorname{PSL}_{2}(\\mathbb{F}_{5})$ ,\n$\\operatorname{PGL}_{2}(\\mathbb{F}_{5})$ , or\n$A_{6}$ . When\n$G$ is isomorphic to\n$\\operatorname{PSL}_{2}(\\mathbb{F}_{5})$ or\n$\\operatorname{PGL}_{2}(\\mathbb{F}_{5})$ , then\n$\\mathbb{C}(x_{1},\\ldots,x_{6})^{G}$ is\n$\\mathbb{C}$ -rational and\n$k(x_{1},\\ldots,x_{6})^{G}$ is stably\n$k$ -rational for any field\n$k$ . The invariant theory of wreath products will be investigated also.