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Register a new userOn pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays
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The group of almost Riordan arrays contains the group of Riordan arrays as a subgroup. In this note, we exhibit examples of pseudo-involutions, involutions and quasi-involutions in the group of almost Riordan arrays.
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If H is a connected, graded Hopf algebra, then Takeuchis formula can be used to compute its antipode. However, there is usually massive cancellation in the result. We show how sign-reversing involutions can sometimes be used to obtain cancellation-free formulas. We apply this idea to nine different examples. We rederive known formulas for the antipodes in the Hopf algebra of polynomials, the shuffle Hopf algebra, the Hopf algebra of quasisymmertic functions in both the monomial and fundamental bases, the Hopf algebra of multi-quasisymmetric functions in the fundamental basis, and the incidence Hopf algebra of graphs. We also find cancellation-free expressions for particular values of the antipode in the immaculate basis for the noncommutative symmetric functions as well as the Malvenuto-Reutenauer and Porier-Reutenauer Hopf algebras, some of which are the first of their kind. We include various conjectures and suggestions for future research.
Let $G=QD_{8k}~$ be the quasi-dihedral group of order $8n$ and $theta$ be an automorphism of $QD_{8k}$ of finite order. The fixed-point set $H$ of $theta$ is defined as $H_{theta}=G^{theta}={xin G mid theta(x)=x}$ and generalized symmetric space $Q$ of $theta$ given by $Q_{theta}={gin G mid g=xtheta(x)^{-1}~mbox{for some}~xin G}.$ The characteristics of the sets $H$ and $Q$ have been calculated. It is shown that for any $H$ and $Q,~~H.Q eq QD_{8k}.$ the $H$-orbits on $Q$ are obtained under different conditions. Moreover, the formula to find the order of $v$-th root of unity in $mathbb{Z}_{2k}$ for $QD_{8k}$ has been calculated. The criteria to find the number of equivalence classes denoted by $C_{4k}$ of the involution automorphism has also been constructed. Finally, the set of twisted involutions $R=R_{theta}={~xin G~mid~theta(x)=x^{-1}}$ has been explored.
After recalling the definition of Zilber fields, and the main conjecture behind them, we prove that Zilber fields of cardinality up to the continuum have involutions, i.e., automorphisms of order two analogous to complex conjugation on (C,exp). Moreover, we also prove that for continuum cardinality there is an involution whose fixed field, as a real closed field, is isomorphic to the field of real numbers, and such that the kernel is exactly 2{pi}iZ, answering a question of Zilber, Kirby, Macintyre and Onshuus. The proof is obtained with an explicit construction of a Zilber field with the required properties. As further applications of this technique, we also classify the exponential subfields of Zilber fields, and we produce some exponential fields with involutions such that the exponential function is order-preserving, or even continuous, and all of the axioms of Zilber fields are satisfied except for the strong exponential-algebraic closure, which gets replaced by some weaker axioms.
In this article we consider the cycle structure of compositions of pairs of involutions in the symmetric group S_n chosen uniformly at random. These can be modeled as modified 2-regular graphs, giving rise to exponential generating functions. A composition of two random involutions in S_n typically has about n^(1/2) cycles, and the cycles are characteristically of length n^(1/2). Compositions of two random fixed-point-free involutions, on the other hand, typically have about log n cycles and are closely related to permutations with all cycle lengths even. The number of factorizations of a random permutation into two involutions appears to be asymptotically lognormally distributed, which we prove for a closely related probabilistic model. This study is motivated by the observation that the number of involutions in [n] is (n!)^(1/2) times a subexponential factor; more generally the number of permutations with all cycle lengths in a finite set S is n!^(1-1/m) times a subexponential factor, and the typical number of k-cycles is nearly n^(k/m)/k. Connections to pattern avoidance in involutions are also considered.
We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincare maps of measured foliations a suitable definition of return words which yields that the set of first return words to a given word is a symmetric basis of the free group on the underlying alphabet $A$. The set of first return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$.
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