Symmetric extensions of dihedral quandles and triple points of non-orientable surfaces

Quandles with involutions that satisfy certain conditions, called good involutions, can be used to color non-orientable surface-knots. We use subgroups of signed permutation matrices to construct non-trivial good involutions on extensions of odd order dihedral quandles.For the smallest example of order 6 that is an extension of the three-element dihedral quandle R₃, various symmetric quandle homology groups are computed, and applications to the minimal triple point number of surface-knots are given.     

Restricted colored permutations and Chebyshev polynomials

Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating function for the colored permutations which avoid this distinguished set and any set of additional patterns, which we use to find a new set of signed permutations counted by the Catalan numbers and a new set of signed permutations counted by the large Schröder numbers. We go on to use this result to compute the generating functions for colored permutations which avoid our distinguished set and any layered permutation with three or fewer layers. We express these generating functions in terms of Chebyshev polynomials of the second kind and we show that they are special cases of generating functions for involutions which avoid 3412 and a layered permutation.     

Note on enumeration of partitions contained in a given shape

Carlitz, Handa, and Mohanty proved determinantal formulas for counting partitions contained in a fixed bounding shape by area. Gessel and Viennot introduced a combinatorial method for proving such formulas by interpreting the determinants as counting suitable configurations of signed lattice paths. This note describes an alternative combinatorial approach that uses sign-reversing involutions to prove matrix inversion results. Combining these results with the classical adjoint formula for the inverse of a matrix, we obtain a new derivation of the Handa–Mohanty determinantal formula.     

Pattern avoidance and Boolean elements in the Bruhat order on involutions

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].     

Restricted Symmetric Permutations

Pattern-avoiding involutions, which have received much enumerative attention, are pattern-avoiding permutations which are invariant under the natural action of a certain subgroup of D ₈, the symmetry group of a square. Three other nontrivial subgroups of D ₈ also have invariant permutations under this action. For each of these subgroups, we enumerate the set of permutations which are invariant under the action of the subgroup and which also avoid a given set of forbidden patterns. The sets of forbidden patterns we consider include all subsets of S ₃. For each subgroup we also give a bijection between the invariant permutations and certain symmetric signed permutations. Received September 14, 2006     

Strongly Anisotropic Involutions on Central Simple Algebras

The classical theorem of Bröcker and Prestel on quadratic forms over formally real fields determines a valuation theoretic condition under which all totally indefinite forms are weakly isotropic. In this article, we look for analogues of such a result in a more general setting of algebras with involutions. We prove that for involutions of first kind over central simple algebras of index two, one indeed has a Bröcker–Prestel like statement. The connection between two conditions, namely, total indefiniteness and weak isotropy is made via so called gauge functions on central simple algebras.     

Factorization of involutions in characteristic two orthogonal groups: An application of the Jordan form to group theory

This paper uses the theory of the Jordan canonical form for a matrix and the theory of orthogonal sums of isometries in metric vector spaces (quadratic spaces) in order to prove a theorem on the factorization of involutions in the orthogonal groups of metric vector spaces over fields of characteristic two. Using this theorem, a classification scheme for such involutions is devised. This scheme is similar to the scheme for involutions when the field is of characteristic not equal of two.     

Involutions of a Clifford Algebra Induced by Involutions of Orthogonal Group in Characteristic 2

Among the involutions of a Clifford algebra, those induced by the involutions of the orthogonal group are the most natural ones. In this work, several basic properties of these involutions, such as the relations between their invariants, their occurrences, and their decompositions, are investigated.     

Classification of Involutions on Incidence Algebras

For the case when a finite partially ordered set X has an element that is comparable with all the elements of X, we present necessary and sufficient conditions for two involutions on the incidence algebra of X (over a field of characteristic different from 2) to be equivalent via inner automorphism. Furthermore, we use this classification to get the general classification of involutions.     

Calabi–Yau threefolds with non-Gorenstein involutions

The concept of non-Gorenstein involutions on Calabi–Yau threefolds is a higher dimensional generalization of non-symplectic involutions on K3 surfaces. We present some elementary facts about Calabi–Yau threefolds with non-Gorenstein involutions. We give a classification of the Calabi–Yau threefolds of Picard rank one with non-Gorenstein involutions, whose fixed locus is not zero-dimensional.     

Green’s Functions of Partial Differential Equations with Involutions

In this paper we develop a way of obtaining Green''s functions of partial differential equations with linear involutions by reducing the equation to a higher-order PDE without involutions. The developed theory is applied to a model of heat transfer in a conducting plate which is bent in half.     

Counting occurrences of 231 in an involution

We study the generating function for the number of involutions on n letters containing exactly r?0 occurrences of 231. It is shown that finding this function for a given r amounts to a routine check of all involutions of length at most 2r+2.     

Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field. Received: 29 April 1999     

Isotropy of unitary involutions

We prove the so-called unitary isotropy theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.     

On <Emphasis Type="Italic">pq</Emphasis>-hyperelliptic Klein surfaces

In this paper we study compact Klein surfaces of algebraic genus d > 1 admitting p- and q-hyperelliptic involutions by which we mean involutions with the orbit spaces having algebraic genera p and q. We give necessary and sufficient conditions for p, q and d to exist such surfaces. It turns out that these conditions are also sufficient for the existence of such surfaces with commuting involutions what allow us to study this class also. We study the spectrum of hyperellipticity degrees of the product of these involutions and topological type of these surfaces. G. Gromadzki was supported by the grant SAB 2005-0049 of the Spanish Ministry of Education and Sciences. E. Tyszkowska was supported by BW 5100-5-0198-6.     

实厄米Banach*-代数

对任意实厄米的B anach*-代数进行系统研究,即对不需任何附加条件限制的代数进行研究,讨论方法与传统的假定代数具有单位元和*-运算是连续的两个重要假定条件下的方法不同.     

Self-inverse Sheffer sequences and Riordan involutions

In this short note, we focus on self-inverse Sheffer sequences and involutions in the Riordan group. We translate the results of Brown and Kuczma on self-inverse sequences of Sheffer polynomials to describe all involutions in the Riordan group.     

Discriminants of orthogonal involutions on central simple algebras with tame gauges

In this article we generalize the results of Chacron et al. (1995) [4] concerning the computation of the discriminants of involutions of the first kind on central division algebras over Henselian valued fields of residue characteristic different from 2. We prove analogous results for orthogonal involutions on tame central simple algebras with simple residue algebras over a wide class of Henselian valued fields of arbitrary residue characteristic (see Corollary 2.18).     

A note on inverses of cyclotomic mapping permutation polynomials over finite fields

In this note, we give a shorter proof of the result of Zheng, Yu, and Pei on the explicit formula of inverses of generalized cyclotomic permutation polynomials over finite fields. Moreover, we characterize all these cyclotomic permutation polynomials that are involutions. Our results provide a fast algorithm (only modular operations are involved) to generate many classes of generalized cyclotomic permutation polynomials, their inverses, and involutions.     

Isotropy of symplectic involutions

We prove the symplectic analogue of the isotropy theorem for orthogonal involutions. We apply (a modification of) a method due to J.-P. Tignol originally applied to prove the symplectic analogue of the hyperbolicity theorem for orthogonal involutions.