Coset Decomposition Research Articles

Elliptic Hecke algebras and modified Cherednik algebras

The elliptic Hecke algebras associated to the 1-codimensional elliptic root systems have been defined by H. Yamada [10], which are subalgebras of Cherednik's double affine Hecke algebras [2, 3]. The elliptic Hecke algebras associated to the elliptic root systems of type $X^{(1,1)}$ have been defined similarly by the author [11] in terms of generators and relations associated to the completed elliptic diagram. On the other hand, M. Kapranov [6] has defined modified Cherednik algebras associated to the double coset decomposition of the group schemes over 2-dimensional local field. In this paper, we see that modified Cherednik algebras are isomorphic to elliptic Hecke algebras of type $X^{(1,1)}$.

Coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids

We obtain several new results about the coding of a monoid by an appropriate submonoid of a polycyclic monoid. In particular, we characterize groups, periodic monoids, and right cancellative monoids in terms of the coset decomposition of positively self-conjugate inverse submonoids of polycyclic monoids.

Hecke Algebras and Automorphic Forms

The goal of this paper is to carry out some explicit calculations of the actions of Hecke operators on spaces of algebraic modular forms on certain simple groups. In order to do this, we give the coset decomposition for the supports of these operators. We present the results of our calculations along with interpretations concerning the lifting of forms. The data we have obtained is of interest both from the point of view of number theory and of representation theory. For example, our data, together with a conjecture of Gross, predicts the existence of a Galois extension of Q with Galois group G2(F5) which is ramified only at the prime 5. We also provide evidence of the existence of the symmetric cube lifting from PGL2 to PGSp4.

Double coset decompositions and intertwining

Let F be a non-archimedean local field of residual characteristic different from 2. This paper is a first step in the description of the smooth representation theory of a unitary group G ? GL(N,F) via types. We intersect certain double cosets in GL(N,F) with G and hence obtain the intertwining of certain characters ?s- of open compact subgroups of G, for s ? Lie G. In the case when s is elliptic regular "modulo &weierpF", we will then obtain supercuspidal representations of G.

Coset and double-coset decompositions of the magnetic point groups.

The coset and double-coset decompositions of the 420 subgroups of m(z)3(xyz)m(xy)1' (O(h)1') and the 236 subgroups of 6(z)/m(z)m(x)m(1)1' (D(6h)1') with respect to each of their subgroups are calculated along with additional mathematical properties of these groups.

Double coset decompositions and computational harmonic analysis on groups

In this paper we introduce new techniques for the efficient computation of a Fourier transform on a finite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorithms that generalize the Cooley-Tukey FFT to arbitrary finite group. We apply our general results to special linear groups and low rank symmetric groups, and obtain new efficient algorithms for harmonic analysis on these classes of groups, as well as the two-sphere.

Minimal and Maximal Semigroups Related to Causal Symmetric Spaces

be an irreducible globally hyperbolic semisimple symmetric space, and let S⊆G be a subsemigroup containing H not isolated in S. We show that if So≠ 0 then there are H-invariant minimal and maximal cones Cmin⊆Cmax in the tangent space at the origin such that H exp Cmin⊆S⊆HZK(a)expCmax. A double coset decomposition of the group G in terms of Cartan subspaces and the group H is proved. We also discuss the case where G/H is of Cayley type.

Decomposition of the Moonshine Vertex Operator Algebra as Virasoro Modules

In this article, we obtain a decomposition of the Moonshine vertex operator algebra V♮ associated with the algebra (L(1/2,0)⊗L(7/10,0)⊗L(4/5,0))⊗12. Our method is based on a coset decomposition of the Leech lattice Λ associated with 2A212 using some codes. In fact, we construct a code vertex operator algebra MkC⊗MtD with a ternary code D and a Z2×Z2 code C and use it to obtain a decomposition of V♮.

Double Coset Decompositions of Groups

We show that residually finite or word hyperbolic groups which can be decomposed as a finite union of double cosets of a cyclic subgroup are necessarily virtually cyclic, and we apply this result to the study of Frobenius permutation groups. We show that in general, finite double coset decompositions of a group can be interpreted as an obstruction to splitting a group as a free product with amalgamation or an HNN extension.

Magnetic subperiodic groups.

The magnetic subperiodic groups, the 31 magnetic frieze-group types, the 394 magnetic rod-group types and the 528 magnetic layer-group types, are derived and given symbols based on the symbols for the nonmagnetic subperiodic groups in Volume E of International Tables for Crystallography. The symbols are constructed in analogy to the Opechowski-Guccione symbols for magnetic space groups. Tables are given that list one group from each type. Each group is specified not only by its symbol but also by explicitly listing the coset representatives of the coset decomposition of the group with respect to its translational subgroup.

Recursive fast computation of the two-dimensional discrete cosine transform

An efficient algorithm is presented for computing the two-dimensional discrete cosine transform (2-D DCT) whose size is a power of a prime. Based on a generalised 2-D to one-dimensional (1-D) index mapping scheme, the proposed algorithm decomposes the 2-D DCT outputs into three parts. The first part forms a 2D DCT of a smaller size. The remaining outputs are further decomposed into two parts, depending on the summation of their indices. The latter two parts can be reformulated as a set of circular correlation (CC) or skew-circular correlation (SCC) matrix-vector products by utilising the previously addressed maximum coset decomposition. Such a decomposition procedure can be repetitively carried out for the 2-D DCT of the first part, resulting in a sequence of CC and SCC matrix-vector products of various sizes. Employing fast algorithms for the computation of these CC/SCC operations can substantially reduce the numbers of multiplications compared with those of the conventional row-column decomposition approach. In the special case where the data size is a power of two, the proposed algorithm can be further simplified, calling for computations comparable with those of previous works.

Lp-multipliers of mixed-norm type on locally compact Vilenkin groups

AbstractLet G be a locally compact Vilenkin group with dual group Γ. We prove Littlewood-Paley type inequalities corresponding to arbitrary coset decompositions of Γ. These inequalities are then applied to obtain new Lp(G) multiplier theorems. The sharpness of some of these results is also discussed.

Some new bounds for q-ary linear block codes

In this correspondence, we derive, with the method of coset decompositions, some new bounds for q-ary linear block codes, where q is the order of some finite field.

On classification of special linear group over commutative rings

In [2] and [3] we described the double coset decomposition for the modular group SLr+s (Z) with respect to the congruence subgroup Р r,s (m) for any positive integer m. In the present paper we obtain the double coset decomposition of SLr+s (R) with respect to Р r,s (I). Where R is a commutative ring with unioty 1, and I is a nonzero ideal in R such that R/I is the direct sum of local principal ideal rings; thus generalize the results obtained in [2] and [3].

Double Coset Decompositions of Reductive Lie Groups Arising from Two Involutions

(G)0 ⊂ H ⊂ G and (G )0 ⊂ L ⊂ G . Here G = {g ∈ G | ρ(g) = g} for an automorphism ρ of G and F0 denote the connected component of F containing the identity e for a Lie group F . In [8], we gave standard representatives of double coset decompositions H\G/L for some typical triples (G,H,L) by only using “algebraic” method. As the most elementary example, we studied the case (G,H,L) = (GL(n, IF), GL(p, IF)×GL(n−p, IF), GL(r, IF)× GL(n − r, IF)) for an arbitrary field IF. We also studied in [8] other examples related to quadratic forms on vector spaces over the fields IR,C and IH. In this paper, we assume that G is a reductive Lie group and we will describe the structure of the double coset decomposition H\G/L for an arbitrary (G,H,L). First we study the compact case (Section 3) and next the noncompact case (Section 4). In Section 3, we assume that the semisimple part Gs of G is compact. (Gs is the analytic subgroup of G for gs = [g, g].) When G = Gs and

Non-magnetic twin laws

Twin laws are groups that express the symmetry relationships between two simultaneously observed domain states (domain pair) and are used to determine physical properties that can distinguish between the observed domains. A tabulation is presented of all possible non-magnetic twin laws, that is, all possible symmetry groups and twinning groups of the domain pair. Additional information is provided related to determining twin laws. This includes the coset and double-coset decomposition of point groups, the indexing and point-group symmetry of domain states, permutations of domain states, and a classification of domain states.

Double Coset Decompositions of Algebraic Groups Arising from Two Involutions I

Let G be a real reductive algebraic group with two involutive automorphisms (involutions) σ and τ (σ = τ 2 = id.). Put H = G and L = G where G = {x ∈ G | ρx = x} for an automorphism ρ of G. If G is connected and one of the following three conditions is satisfied, then the structure of the double coset decomposition H\G/L is studied. (a) G is compact and στ = τσ. (b) σ or τ is a Cartan involution. (c) σ = τ . In (b), we may assume that στ = τσ ([1], [9]). If (a) or (b) is satisfied, then we can choose representatives of H\G/L in a maximal connected abelian subgroup A contained in {g ∈ G | σ(g) = τ(g) = g−1} ([4], [5], [9]). Consider the case (c). By the map φ(g) = gσ(g)−1, the double coset decomposition H\G/H is identified with the decomposition of φ(G) into H-cojugacy classes. It is easy to see that the set φ(G) is closed under the Jordan decomposition. (Let x = xsxu be the Jordan decomposition of x ∈ φ(G). Then xs, xu ∈ φ(G). c.f. [14]) The structure of semisimple elements in φ(G) is studied in [14] by generalizing the notion of Cartan subgroups. Let q be the subspace of the Lie algebra g of G defined by q = {X ∈ g | σX = −X}. Then the exponential map gives a bijection between the nilpotent H-conjugacy classes in q and the unipotent H-conjugacy classes in φ(G). Nilpotent H-conjugacy classes in q (and their singularities, closure relations etc.) are studied in several papers ([7],[12],[13],[11],[16] and [17]). Especially, suppose that G = G1 × G1 and σ(x, y) = (y, x) for x, y ∈ G1. Then H = G = {(x, x) | x ∈ G1}. Since G/H is identified with G1 by the map (x, y)H 7→ xy−1, the double coset decomposition H\G/H is identified with the decomposition of G1 into conjugacy classes.

Two-stage decomposition of the DCT

The DCT kernel matrix is first decomposed into a block diagonal structure (BDS) with diagonal skew-circular correlated (SCCR) sub-matrices of length 2, 4, ..., N/2 by coset decomposition, and then each of these independent SCCR sub-matrices is further split into two stages by decomposing its elements into a linear combination of other simple basis functions. The preprocessing stage can be treated as a new transform approximated to the DCT, and is suitable for image compression. Various preprocessing stages are obtained by choosing various basis functions. The postprocessing stage is used for converting the preprocessing stage back to the DCT. Both the preprocessing stage and the postprocessing stage are BDSs containing independent diagonal SCCR sub-matrices, thus the fast and parallel computation of both the preprocessing and the postprocessing stages is feasible using methods such as a semisystolic array or distributed arithmetic implementation.

Skew-circular/circular correlation decomposition of prime-factor DCT

An algorithm to decompose the prime-factor DCT into skew-circular/circular correlation (SCC/CC) by coset decomposition is proposed. The simplest case is when the two factors are odd and relatively prime. In this case, the DCT output components are split into six subgroups. Each subgroup contains three short-length CC or SCC matrix-vector products, and the three products can further be merged into only one short-length SCC or CC matrix-vector product. The six subgroups are independent, thus parallel computing is feasible. By fast computation of the short-length SCC and CC, this algorithm reaches the same or less number of multiplications compared to other efficient prime-factor algorithms.

A diagram technique for nonorthogonal electron group functions. I. Right coset decomposition of symmetric group

For an N-electron system partitioned into a set of electron groups, an expanded arrow diagram (EAD) technique is proposed. It is useful when the wave function of the system is approximated by an antisymmetrized product of mutually nonorthogonal group functions. The technique provides a graphic representation of the right (left) coset decomposition of the symmetric group SN with respect to its subgroup S0 generated by all the intragroup permutations. The connection between the EADs and arrow diagrams of Matsen, Klein et al., and Wilson is stated and used to prove the validity of the arrow diagram representation of the double coset decomposition of SN with respect to S0, as well as to obtain the decomposition coefficients explicitly. The technique is applicable to systems composed of any finite or infinite number of electron groups each containing an arbitrary number of electrons.