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Hiroyoshi Yamaki 《Proceedings of the American Mathematical Society》2008,136(2):397-402
We will give an estimation of the order of a group of even order by the order of the centralizer of an involution using the classification of finite simple groups.
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Axel Hultman 《Advances in Mathematics》2005,195(1):283-296
Applying a classical theorem of Smith, we show that the poset property of being Gorenstein* over Z2 is inherited by the subposet of fixed points under an involutive poset automorphism. As an application, we prove that every interval in the Bruhat order on (twisted) involutions in an arbitrary Coxeter group has this property, and we find the rank function. This implies results conjectured by F. Incitti. We also show that the Bruhat order on the fixed points of an involutive automorphism induced by a Coxeter graph automorphism is isomorphic to the Bruhat order on the fixed subgroup viewed as a Coxeter group in its own right. 相似文献
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Mireille?Bousquet-MélouEmail author Einar?Steingrímsson 《Journal of Algebraic Combinatorics》2005,22(4):383-409
In a recent paper, Backelin, West and Xin describe a map φ* that recursively replaces all occurrences of the pattern k... 21 in a permutation σ by occurrences of the pattern (k−1)... 21 k. The resulting permutation φ*(σ) contains no decreasing subsequence of length k. We prove that, rather unexpectedly, the map φ* commutes with taking the inverse of a permutation.
In the BWX paper, the definition of φ* is actually extended to full rook placements on a Ferrers board (the permutations correspond to square boards), and the construction
of the map φ* is the key step in proving the following result. Let T be a set of patterns starting with the prefix 12... k. Let T′ be the set of patterns obtained by replacing this prefix by k... 21 in every pattern of T. Then for all n, the number of permutations of the symmetric group
n that avoid T equals the number of permutations of
n that avoid T′.
Our commutation result, generalized to Ferrers boards, implies that the number of involutions of
n that avoid T is equal to the number of involutions of
n avoiding T′, as recently conjectured by Jaggard.
Both authors were partially supported by the European Commission's IHRP Programme, grant HPRN-CT-2001-00272, “Algebraic Combinatorics
in Europe” 相似文献
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Shigeo Koshitani 《代数通讯》2013,41(10):4308-4321
We determine all finite groups G such that the Loewy length (socle length) of the projective cover P(k G ) of the trivial kG-module k G is four, where k is a field of characteristic p > 0 and kG is the group algebra of G over k, by using previous results and also the classification of finite simple groups. As a by-product we prove also that if p = 2 then all finite groups G such that the Loewy lengths of the principal block algebras of kG are four, are determined. 相似文献
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We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this
property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin
paths is demonstrated.
This article is largely based on results from the second author’s M.Sc. thesis [15]. 相似文献
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R.E. Stong 《Topology and its Applications》1985,19(2):169-188
Being given a closed manifold Mn, there are involutions (X2n, T) on closed manifolds of twice the dimension having fixed point set M. Kulkarni defined the deficiency of M for a class of involutions to be for all involutions (X, T) in the class. This paper exhibits manifolds for which the deficiency is positive for all involutions and studies the deficiencies for other classes. 相似文献
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Hiroyuki Ishibashi 《Czechoslovak Mathematical Journal》2006,56(2):533-541
We define a linear map called a semiinvolution as a generalization of an involution, and show that any nilpotent linear endomorphism
is a product of an involution and a semiinvolution. We also give a new proof for Djocović’s theorem on a product of two involutions. 相似文献
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A. M. Popov 《Algebra and Logic》2003,42(2):130-135
We settle Question 10.61, posed by A. Sozutov in the Kourovka Notebook, for the case where
. 相似文献