Complete Set Of Representatives Research Articles

Construction of self-dual matrix codes

Matrix codes over a finite field $${\mathbb {F}}_q$$ are linear codes defined as subspaces of the vector space of $$m \times n$$ matrices over $${\mathbb {F}}_q$$ . In this paper, we show how to obtain self-dual matrix codes from a self-dual matrix code of smaller size using a method we call the building-up construction. We show that every self-dual matrix code can be constructed using this building-up construction. Using this, we classify, that is, we find a complete set of representatives for the equivalence classes of self-dual matrix codes of small sizes. In particular we have classifications for self-dual matrix codes of sizes $$2 \times 4$$ , $$2 \times 5$$ over $${\mathbb {F}}_{2}$$ , of size $$2 \times 3$$ , $$2 \times 4$$ over $${\mathbb {F}}_{4}$$ , of size $$2 \times 2$$ , $$2 \times 3$$ over $${\mathbb {F}}_{8}$$ , and of size $$2 \times 2$$ , $$2 \times 3$$ over $${\mathbb {F}}_{13}$$ , all of which have been left open from K. Morrison’s classification.

On the control of fusion in the local category for the p-block with a minimal nonabelian defect group

If B is a p-block of a finite group G with a minimal nonabelian defect group D(p is an odd prime number) and (D, bD) is a Sylow B-subpair of G, then NG(D, bD) controls B-fusion of G in most cases. This result is of great importance, because we can use it to obtain a complete set of representatives of G-conjugate classes of B-subsections and to calculate the number of ordinary irreducible characters in B. This result is key to the calculation of the structure invariants of the block with a minimal nonablian defect group. On the other hand, we improve Brauer’s famous formula $$ k\left( B \right) = \sum\limits_{\left( {\omega ,b_\omega } \right)} {l\left( {b_\omega } \right)} , $$ where (ω, bω) ∈ [(G: sp(B))]. Let p be any prime number, B be a p-block of a finite group G and (D, bD) be a Sylow B-subpair of G. H is a subgroup of NG(D, bD) satisfying NG(R, bR) = NH(R, bR)CG(R), ∀(R, bR) ∈ A0(D, bD) NG(〈ω′〉, bω′) = NH(〈ω′〉, bω′)CG(ω′), ∀(ω′, bω′) ∈ (D, bD)

Path realization of crystal B(∞)

A class of piecewise linear paths, as a generalization of Littelmann’s paths, are introduced, and some operators, acting on the above paths with fixed parametrization, are defined. These operators induce the ordinary Littelmann’s root operators’ action on the equi-alence classes of paths. With these induced operators, an explicit realization of B(∞) is given in terms of equivalence classes of paths, where B(∞) is the crystal base of the negative part of a quantum group U q (g). Furthermore, we conjecture that there is a complete set of representatives for the above model by fixing a parametrization, and we prove the case when g is of finite type.

The Structure of the Reduced Product of Preinjective Modules Over a Hereditary Right Pure Semisimple Ring

ABSTRACT Let G be a division ring and F ⊂ G a fixed subdivision ring of G such that dim F G = ∞, dim G F = 2 and the F-G-bimodule F G G has the infinite dimension type d−∞( F G G ) = (…, 2, 2…,2, 2, 1, ∞) in the sense of Simson (19962000), see Section 2. In particular, the hereditary ring is a counter-example to the pure semisimplicity conjecture. Let be a complete set of representatives of pairwise non-isomorphic preinjective right R G -modules. We prove that the reduced product is a direct sum , where , , s 0 = s 1 = (card G)ℵ0 , and means the direct sum of s j copies of the right ideal P j of R G . We also show that is isomorphic to ∏ℵ0 P 1. Communicated by J. Gomaz Pardo.

Local Regularity of L∞ ∞-Refinable Function Vectors

This article discusses the local regularity of refinable function vectors associated with a dilation matrix M. Suppose that D is a complete set of representatives of Zs/MZs. Under the assumptions that the self-affine tile T (M,D), associated with dilation matrix M and digit set D, has measure 1 and that the corresponding refinable function vector φ ∈ L∞, we prove that there is a set H ⫅ Rs of full measure such that the restriction φ|H of φ on H has a positive Holder exponent α(x) at every x ∈ H. Similar result holds for the derivatives of refinable function vectors provided that the dilation matrix is diagonalizable.

A note on well-generated Boolean algebras in models satisfying Martin's axiom

A Boolean algebra B is well generated, if it has a well-founded sublattice L such that L generates B. Let B be a superatomic Boolean algebra. The rank of B ( rk ( B ) ) is defined to be the Cantor Bendixon rank of the Stone space X of B. For every i ⩽ rk ( B ) let λ i ( B ) be the number of isolated points in the i's Cantor Bendixon derivative of X. The cardinal sequence of B is defined as λ ⇒ ( B ) ≔ 〈 λ i ( B ) : i ⩽ rk ( X ) 〉 . If a ∈ B - { 0 } , then the rank of a ( rk ( a ) ) is defined as the rank of the Boolean algebra B ↾ a ≔ { b ∈ B : b ⩽ a } . An element a ∈ B - { 0 } is a generalized atom ( a ∈ At ^ ( B ) ), if the last cardinal in the cardinal sequence of B ↾ a is 1. Let a , b ∈ At ^ ( B ) . We denote a ∼ B b , if rk ( a ) = rk ( b ) = rk ( a · b ) . A subset H ⊆ At ^ ( B ) is a complete set of representatives ( CSR) for B, if for every a ∈ At ^ ( B ) there is a unique b ∈ H such that b ∼ B a . Any CSR for B generates B. We say that B is hereditarily decreasingly canonically well generated, if for every subalgebra C of B and every CSR H for C there is a CSR M for C such that: (1) for every a ∈ H and b ∈ M : if b ∼ C a then b ⩽ a ; (2) the sublattice of C generated by M is well founded. Theorem. Assume ( MA + ℵ 1 < 2 ℵ 0 ) . Let ε be a countable ordinal, κ < 2 ℵ 0 and n < ω . If B is a superatomic Boolean algebra such that λ ⇒ ( B ) = 〈 ℵ 0 : i < ε 〉 ^ 〈 κ , ℵ 1 , n 〉 or λ ⇒ ( B ) = 〈 ℵ 0 , 2 ℵ 0 , ℵ 1 , n 〉 , then B is hereditarily decreasingly canonically well generated.

On essentially low, canonically well-generated Boolean algebras

AbstractLet B be a superatomic Boolean algebra (BA). The rank of B (rk(B)). is defined to be the Cantor Bendixon rank of the Stone space of B. If a ∈ B − {0}, then the rank of a in B (rk(a)). is defined to be the rank of the Boolean algebra . The rank of 0B is defined to be −1. An element a ∈ B − {0} is a generalized atom , if the last nonzero cardinal in the cardinal sequence of B ↾ a is 1. Let a, b ∈ . We denote a ˜ b, if rk(a) = rk(b) = rk(a · b). A subset H ⊆ is a complete set of representatives (CSR) for B, if for every a there is a unique h ∈ H such that h ~ a. Any CSR for B generates B. We say that B is canonically well-generated (CWG), if it has a CSR H such that the sublattice of B generated by H is well-founded. We say that B is well-generated, if it has a well-founded sublattice L such that L generates B.Theorem 1. Let B be a Boolean algebra with cardinal sequence . If B is CWG, then every subalgebra of B is CWG.A superatomic Boolean algebra B is essentially low (ESL), if it has a countable ideal I such that rk(B/I) ≤ 1.Theorem 1 follows from Theorem 2.9. which is the main result of this work. For an ESL BA B we define a set FB of partial functions from a certain countably infinite set to ω (Definition 2.8). Theorem 2.9 says that if B is an ESL Boolean algebra, then the following are equivalent.(1) Every subalgebra of B is CWG: and(2) FB is bounded.Theorem 2. If an ESL Boolean algebra is not CWG, then it has a subalgebra which is not well-generated.

Unipotent Orbital Integrals of Hecke Functions for GL(n)

AbstractLet G = GL(n, F) where F is a p-adic field, and let 𝓗(G) denote the Hecke algebra of spherical functions on G. Let u1,..., up denote a complete set of representatives for the unipotent conjugacy classes in G. For each 1 ≤ i ≤ p, let μi be the linear functional on such that μi(f) is the orbital integral of f over the orbit of ui. Waldspurger proved that the μi, 1 ≤ i ≤ p, are linearly independent. In this paper we give an elementary proof of Waldspurger's theorem which provides concrete information about the Hecke functions needed to separate orbits. We also prove a twisted version of Waldspurger's theorem and discuss the consequences for SL(n, F).

A characterization of blocks with vertices

In modular representation theory, one regards a block B of a finite group G as a G-algebra or a G x G-module. In [6], Green defined G-algebras and generalized defect groups to G-algebras, and proved that the G x G-module B have vertex DA = {(d, d) E G x G: LIE Dj, where D is the defect group of B. After, in [7, Sect. 31, Puig introduced the concept of interior G-algebras which is a special case of G-algebras. An interior G-algebra becomes a G-algebra and a G x G-module. In this paper, we treat mainly interior G-algebras. Let G be a finite group, p a prime number and k a splitting field for G of characteristic p. Let 0 be a complete discrete valuation ring with unique maximal ideal (rr) and the residue field O/(z) is k. In this paper, any Co-algebra A is an O-free module with a finite rank as an O-module, and any A-module is an O-free left A-module with an finite rank as an Comodule. For an O-algebra A and O-algebra homomorphism p of O[G] to A such that p( 1) = 1 A, where 1 A is the identity element of A, the pair (A, p) is called an interior G-algebra. Then the O-algebra A becomes a G x Gmodule defined by (g, h) a = p(g) ap(h ‘), where a is an element of A and (g, h) is an element of G x G. Let V be an Co[G]-module and H be a subgroup of G. A subspace VH consists of the fixed points of V under the action of H. Whenever H’ is a subgroup of G such that H < H’, the trace map Tr;‘: VH + V“’ is defined by v H C gv, where g runs over a complete set of representatives in H’ of H’/H. We denote by Vz’ the image Trz’ (V”). For p-subgroup P of G we denote by V(P) the O-module VP/C Vg + (rc) VP, where Q runs over the family of proper subgroups of P. See [2] or [3]. For an interior G-algebra (A, p) we often apply this notation to the O[G*]-module AIGd, which is the restriction of A to Gd = {(g, g) E G x G: g E G}, and use similar notation AH, Tr:‘, AZ’ and A(P). Then AZ’ is a two-sided ideal in AH’. If AC is a local ring, we call the 344 OOZl-8693/87 $3.00

Varieties and transfers

be a minimal KG-projective resolution of the trivial kG-module k. The restriction of (P, E) to H is a kH-projective resolution. Any <e H;(k) is represented by a kHcocycle f E Horn&P,, k). The transfer to G of i is then Trz([)=cIsCfl) where f’(m) = xi= 1 f(xip ‘m) for x t, . . . ,x, a complete set of representatives of the left cosets of H in G and m E P, . The transfer map is only a k-module homomorphism. Its image in Hz(k) is an ideal. The transfer has been studied for many years. However when p divides IG: HI, its utility is somewhat limited because it is often zero. We show here that, collectively, the images of the transfers are relatively large provided the Sylow p-subgroup of G is not commutative. Specifically we prove the following.

The Bordism Ring of Manifolds with Involution

A complete set of representatives is found for the bordism ring of manifolds with involution. The multiplicative structure is expressed in terms of these representatives, and in particular, a set of ring generators can be isolated. Also the constructions involved are used to give an insight into a map of J. M. Boardman.

The bordism ring of manifolds with involution

A complete set of representatives is found for the bordism ring of manifolds with involution. The multiplicative structure is expressed in terms of these representatives, and in particular, a set of ring generators can be isolated. Also the constructions involved are used to give an insight into a map of J. M. Boardman.

On the decomposition of cyclic codes into cyclic classes

The problem of finding the cycle representatives of a cyclic code is examined. For an ( n, k ) cyclic code whose parity check polynomial h(x) consists entirely of factors with exponent n , a complete set of representatives is obtained. Further, if the exponents of the factors of h(x) are relatively prime to each other, a complete set can also be generated. For other cyclic codes, a large number of representatives can also be found and relations are described which assist in finding the remaining ones. Some results are presented for counting the number of cycles of specified length. The relation of this work to the weight distribution problem and word synchronization of cyclic codes is mentioned.