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1.
In this article, we consider indecomposable Specht modules with abelian vertices. We show that the corresponding partitions are necessarily p 2-cores where p is the characteristic of the underlying field. Furthermore, in the case of p≥3, or p=2 and μ is 2-regular, we show that the complexity of the Specht module S μ is precisely the p-weight of the partition μ. In the latter case, we classify Specht modules with abelian vertices. For some applications of the above results, we extend a result of M. Wildon and compute the vertices of the Specht module S(pp)S^{(p^{p})} for p≥3.  相似文献   

2.
During the 2004–2005 academic year the VIGRE Algebra Research Group at the University of Georgia (UGA VIGRE) computed the complexities of certain Specht modules S λ for the symmetric group Σ d , using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The UGA VIGRE Algebra Group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of λ is built out of p×p blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the p-weight of partitions and branching.  相似文献   

3.
Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module $S^\lambda$ has non-trivial cyclic vertex if and only if $\lambda$ has p-weight 1. 2. If p does not divide n and $S^{(n-r, 1^r)}$ is indecomposable then its vertex is a p-Sylow subgroup of $S_{n-r-1} \times S_r$.Received: 15 August 2002  相似文献   

4.
We describe a particularly easy way of evaluating the modular irreducible matrix representations of the symmetric group. It shows that Specht’s approach to the ordinary irreducible representations, along Specht polynomials, can be unified with Clausen’s approach to the modular irreducible representations using symmetrized standard bideterminants. The unified method, using symmetrized Specht polynomials, is very easy to explain, and it follows directly from Clausen’s theorem by replacing the indeterminate xij of the letter place algebra by xji.Our approach is implemented in SYMMETRICA. It was used in order to obtain computational results on code theoretic properties of the p-modular irreducible representation [λ]p corresponding to a p-regular partition λ via embedding it into representation spaces obtained from ordinary irreducible representations. The first embedding is into the permutation representation induced from the column group of a standard Young tableau of shape λ. The second embedding is the embedding of [λ]p into the space of , the p-modular representation obtained from the ordinary irreducible representation [λ] by reducing the coefficients modulo p.We include a few tables with dimensions and minimum distances of these codes; others can be found via our home page.  相似文献   

5.
David J. Hemmer 《代数通讯》2013,41(11):3292-3306
The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for GL n (k). This article is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology.

Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for GL n (k), these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of GL n (k)-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be indecomposable self dual trivial source modules.  相似文献   

6.
Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dimp(A) £ s-crs{{\rm {dim}_p}(A)\le s-c\varrho^s} where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dimp(A) £ dimp(X)-c(log\tfrac1r)-1rt{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .  相似文献   

7.
We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ-measurable operators E(M,t){E(\mathcal{M},\tau)} associated to a semifinite von Neumann algebra (M, t){(\mathcal{M}, \tau)}. We prove that x is a smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f) = 0, for the function f ? SE×{f\in S_{E^{\times}}} supporting μ(x), or s(x *) = 1. Under the assumption that the trace τ on M{\mathcal{M}} is σ-finite, we show that x is strongly smooth point of the unit ball in E(M, t){E(\mathcal{M}, \tau)} if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ(f). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M,t){E(\mathcal{M},\tau)}.  相似文献   

8.
Sinéad Lyle 《代数通讯》2013,41(5):1723-1752
We determine the v-decomposition numbers d μλ(v) for μ a partition with at most three parts. We use this information to compute the composition factors of the Specht modules of the Hecke algebra ?0 = ??, ω( n ) which correspond to partitions with at most three parts.  相似文献   

9.
Let Λ={λ 1⋅⋅⋅λ s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ i nodes in the ith row. Let λ j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ i +λ j ′−ij+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.   相似文献   

10.
Let Σn be the symmetric group on n letters. For l ≤ n identify Σl with a subgroup of Σn in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣnΣl that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣn. We make a conjecture about how to construct all simple kΣnΣl-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.  相似文献   

11.
A partition of an integer n is a representation n=a 1+a 2+⋅⋅⋅+a k , with integer parts 1≤a 1a 2≤…≤a k . For any fixed positive integer p, a p-succession in a partition is defined to be a pair of adjacent parts such that a i+1a i =p. We find generating functions for the number of partitions of n with no p-successions, as well as for the total number of such successions taken over all partitions of n. In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula for the number of partitions is used to obtain an asymptotic estimate for the average number of p-successions in the partitions of n. This material is based upon work supported by the National Research Foundation under grant number 2053740.  相似文献   

12.
Let (Ω,A,μ) be a probability space, K the scalar field R of real numbers or C of complex numbers,and (S,X) a random normed space over K with base (ω,A,μ). Denote the support of (S,X) by E, namely E is the essential supremum of the set {AA: there exists an element p in S such that X p (ω) > 0 for almost all ω in A}. In this paper, Banach-Alaoglu theorem in a random normed space is first established as follows: The random closed unit ball S *(1) = {fS *: X * f ⩽ 1} of the random conjugate space (S *,X *) of (S,X) is compact under the random weak star topology on (S *,X *) iff EA=: {EA | AA} is essentially purely μ-atomic (namely, there exists a disjoint family {A n : nN} of at most countably many μ-atoms from EA such that E = ∪ n=1 A n and for each element F in EA, there is an H in the σ-algebra generated by {A n : nN} satisfying μ(FΔH) = 0), whose proof forces us to provide a key topological skill, and thus is much more involved than the corresponding classical case. Further, Banach-Bourbaki-Kakutani-Šmulian (briefly, BBKS) theorem in a complete random normed module is established as follows: If (S,X) is a complete random normed module, then the random closed unit ball S(1) = {pS: X p ⩽ 1} of (S,X) is compact under the random weak topology on (S,X) iff both (S,X) is random reflexive and EA is essentially purely μ-atomic. Our recent work shows that the famous classical James theorem still holds for an arbitrary complete random normed module, namely a complete random normed module is random reflexive iff the random norm of an arbitrary almost surely bounded random linear functional on it is attainable on its random closed unit ball, but this paper shows that the classical Banach-Alaoglu theorem and BBKS theorem do not hold universally for complete random normed modules unless they possess extremely simple stratification structure, namely their supports are essentially purely μ-atomic. Combining the James theorem and BBKS theorem in complete random normed modules leads directly to an interesting phenomenum: there exist many famous classical propositions that are mutually equivalent in the case of Banach spaces, some of which remain to be mutually equivalent in the context of arbitrary complete random normed modules, whereas the other of which are no longer equivalent to another in the context of arbitrary complete random normed modules unless the random normed modules in question possess extremely simple stratification structure. Such a phenomenum is, for the first time, discovered in the course of the development of random metric theory.  相似文献   

13.
We give general conditions on a generator of a C0-semigroup (resp. of a C0-resolvent) on Lp(E,μ), p ≥ 1, where E is an arbitrary (Lusin) topological space and μ a σ-finite measure on its Borel σ-algebra, so that it generates a sufficiently regular Markov process on E. We present a general method how these conditions can be checked in many situations. Applications to solve stochastic differential equations on Hilbert space in the sense of a martingale problem are given. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday  相似文献   

14.
It is shown that a K-quasiminimizer u for the one-dimensional p-Dirichlet integral is a K′-quasiminimizer for the q-Dirichlet integral, 1  ≤  q  <  p 1(p, K), where p 1(p, K) > p; the exact value for p 1(p, K) is obtained. The inverse function of a non-constant u is also K′′-quasiminimizer for the s-Dirichlet integral and the range of the exponent s is specified. Connections between quasiminimizers, superminimizers and solutions to obstacle problems are studied.  相似文献   

15.
We realize the integral Specht modules for the symmetric group S n as induced modules from the subalgebra of the group algebra generated by the Jucys–Murphy elements. We deduce from this that the simple modules for ${{\mathbb F}_p} S_n $ are generated by reductions modulo p of the corresponding Jucys–Murphy idempotents.  相似文献   

16.
Let E be a Banach lattice and L1(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L1(μ, E) are given. It is shown that Ls(μ, Z(E)) is the ideal centre of L1(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.   相似文献   

17.

Regarding the Specht modules associated to the two-row partition (n, n), we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the A1-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

  相似文献   

18.
LetG be an infinite compact abelian group,μ a Borel measure onG with spectrumE, and 0<p<1. We show that ifμ is not absolutely continuous with respect to Haar measure, thenL E P (G), the closure inL p (G) of theE-trigonometric polynomials, does not have enough continuous linear functionals to separate points. Ifμ is actually singular, thenL E p (G) does not have any nontrivial continuous linear functionals at all. Our methods recover the classical F. and M. Riesz theorem, and a related several variable result of Bochner; they reveal the existence of small sets of characters that spanL P (T), where T is the unit circle; and they show that theH p spaces of the “big disc algebra” have one-dimensional dual.  相似文献   

19.
The Frattini Subalgebra of Restricted Lie Superalgebras   总被引:6,自引:0,他引:6  
In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra (L, [p]). We show first that if L = A1 + A2 +… +An, then Фp(L) = Фp(A1) +Фp(A2) +…+Фp(An), where each Ai is a p-ideal of L. We then obtain two results: F(L) = Ф(L) = J(L) = L if and only if L is nilpotent; Fp(L) and F(L) are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.  相似文献   

20.
Let Φ be a root system of typeA , ℓ ≧ 2,D , ℓ ≧ 4 orE , 6 ≧ ℓ ≧ 8 andG a group generated by nonidentity abelian subgroupsA r,r∈Φ, satisfying:
(i)  [A r, As]=1 ifs≠−r and ∉ Φ,
(ii)  [A r, As]≦A r+s ifr+s∈Φ,
(iii)  X r=〈Ar, A−r〉 is a rank one group.
Then it is shown, using [3], thatG is a central product of Lie-type groups corresponding to a decomposition of Φ into root-subsystems.  相似文献   

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