Set Of Involutions Research Articles

Zero-sum partitions of Abelian groups and their applications to magic- and antimagic-type labelings

The following problem has been known since the 80s. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $\{m_i\}_{i=1}^{t}$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $\{S_i\}_{i=1}^{t}$ such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $1 \leq i \leq t$. Such a subset partition is called a \textit{zero-sum partition}. $|I(\Gamma)|\neq 1$, where $I(\Gamma)$ is the set of involutions in $\Gamma$, is a necessary condition for the existence of zero-sum partitions. In this paper, we show that the additional condition of $m_i\geq 4$ for every $1 \leq i \leq t$, is sufficient. Moreover, we present some applications of zero-sum partitions to magic- and antimagic-type labelings of graphs.

Generating the level 2 subgroup by involutions

We obtain a minimal generating set of involutions for the level 2 subgroup of the mapping class group of a closed nonorientable surface.

Generating Sets of Conjugate Involutions of Groups PSLn(9)

Generating Sets of Conjugate Involutions of Groups PSLn(9)

Maxima and visibility in involutions

We enumerate involutions according to the joint distribution of left-to-right and right-to-left maxima. From this computation we deduce the distribution of the static “number of visible pairs”, namely, the number of non-adjacent columns in the bargraph representation of a given permutation that are mutually visible to each other. The corresponding generating functions are expressible in terms of formal continued fractions. We obtain the same distribution over the set of involutions avoiding either the pattern 3412 or 4321. The proofs reside on a well-known bijection between involutions and labeled Motzkin paths.

Zero-sum partitions of Abelian groups of order $2^n$

The following problem has been known since the 80's. Let $\Gamma$ be an Abelian group of order $m$ (denoted $|\Gamma|=m$), and let $t$ and $m_i$, $1 \leq i \leq t$, be positive integers such that $\sum_{i=1}^t m_i=m-1$. Determine when $\Gamma^*=\Gamma\setminus\{0\}$, the set of non-zero elements of $\Gamma$, can be partitioned into disjoint subsets $S_i$, $1 \leq i \leq t$, such that $|S_i|=m_i$ and $\sum_{s\in S_i}s=0$ for every $i$, $1 \leq i \leq t$. It is easy to check that $m_i\geq 2$ (for every $i$, $1 \leq i \leq t$) and $|I(\Gamma)|\neq 1$ are necessary conditions for the existence of such partitions, where $I(\Gamma)$ is the set of involutions of $\Gamma$. It was proved that the condition $m_i\geq 2$ is sufficient if and only if $|I(\Gamma)|\in\{0,3\}$. For other groups (i.e., for which $|I(\Gamma)|\neq 3$ and $|I(\Gamma)|>1$), only the case of any group $\Gamma$ with $\Gamma\cong(Z_2)^n$ for some positive integer $n$ has been analyzed completely so far, and it was shown independently by several authors that $m_i\geq 3$ is sufficient in this case. Moreover, recently Cichacz and Tuza proved that, if $|\Gamma|$ is large enough and $|I(\Gamma)|>1$, then $m_i\geq 4$ is sufficient. In this paper we generalize this result for every Abelian group of order $2^n$. Namely, we show that the condition $m_i\geq 3$ is sufficient for $\Gamma$ such that $|I(\Gamma)|>1$ and $|\Gamma|=2^n$, for every positive integer $n$. We also present some applications of this result to graph magic- and anti-magic-type labelings.

Braces whose additive group has a cyclic maximal subgroup

The problem of constructing all the non-degenerate involutive set theoretic solutions of the Yang-Baxter equation recently has been reduced to the problem of describing all the right braces. In particular, the classification of all finite right braces is fundamental in describing all such solutions of the Yang-Baxter equation. Let H be a right brace of order pn, (H, +) ≌ ℤp × ℤpn-1, where n ⩾ 4 and p is odd prime. In this paper we prove Soc(H)≠1 and classify all right braces H such that │Soc(H)│=pn-1.

New simple solutions of the Yang-Baxter equation and solutions associated to simple left braces

Involutive non-degenerate set theoretic solutions of the Yang-Baxter equation are considered, with a focus on finite solutions. A rich class of indecomposable and irretractable solutions is determined and necessary and sufficient conditions are found in order that these solutions are simple. Then a link between simple solutions and simple left braces is established, that allows us to construct more examples of simple solutions. In particular, the results answer some problems stated in the recent paper [13].

Characteristic classes of Borel orbits of square-zero upper-triangular matrices

Anna Melnikov provided a parametrization of Borel orbits in the affine variety of square-zero n×n matrices by the set of involutions in the symmetric group. A related combinatorics leads to a construction a Bott-Samelson type resolution of the orbit closures. This allows to compute cohomological and K-theoretic invariants of the orbits: fundamental classes, Chern-Schwartz-MacPherson classes and motivic Chern classes in torus-equivariant theories. The formulas are given in terms of Demazure-Lusztig operations. The case of square-zero upper-triangular matrices is rich enough to include information about cohomological and K-theoretic classes of the double Borel orbits in Hom(Ck,Cm) for k+m=n. We recall the relation with double Schubert polynomials and show analogous interpretation of Rimányi-Tarasov-Varchenko trigonometric weight function.

Generalizations of parabolic Higgs bundles, real structures, and integrability

We introduce a notion of quasi-antisymmetric Higgs $G$-bundles over curves with marked points. They are endowed with additional structures, which replace the parabolic structures at marked points in the parabolic Higgs bundles. The latter means that the coadjoint orbits are attached to the marked points. The moduli spaces of parabolic Higgs bundles are the phase spaces of complex completely integrable systems. In our case the coadjoint orbits are replaced by the cotangent bundles over some special symmetric spaces in such a way that the moduli space of the modified Higgs bundles are still phase spaces of complex completely integrable systems. We show that the moduli space of the parabolic Higgs bundles is the symplectic quotient of the moduli space of the quasi-antisymmetric Higgs bundle with respect to the action of product of Cartan subgroups. Also, by changing the symmetric spaces we introduce quasi-compact and quasi-normal Higgs bundles. Then the fixed point sets of real involutions acting on their moduli spaces are the phase spaces of real completely integrable systems. Several examples are given including integrable extensions of the ${\rm SL}(2)$ Euler-Arnold top, two-body elliptic Calogero-Moser system and the rational ${\rm SL}(2)$ Gaudin system together with its real reductions.

Presentations of groups with even length relations

We study the properties of groups that have presentations in which the generating set is a fixed set of involutions and all additional relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a factorization with respect to a given parabolic subgroup. Furthermore, we give a counterexample, using a cluster group presentation, which demonstrates that this factorization is not necessarily unique.

Symplectic Geometry of the Koopman Operator

We consider the Koopman operator generated by an invertible transformation of a space with a finite countably additive measure. If the square of this transformation is ergodic, then the orthogonal Koopman operator is a symplectic transformation on the real Hilbert space of square summable functions with zero mean. An infinite set of quadratic invariants of the Koopman operator is specified, which are pairwise in involution with respect to the corresponding symplectic structure. For transformations with a discrete spectrum and a Lebesgue spectrum, these quadratic invariants are functionally independent and form a complete involutive set, which suggests that the Koopman transform is completely integrable.

Generating the twist subgroup by involutions

For a nonorientable surface, the twist subgroup is an index [Formula: see text] subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.

Integral Cayley Graphs over Finite Groups

We prove that the spectrum of a Cayley graph over a finite group with a normal generating set S containing with every its element s all generators of the cyclic group 〈s〉 is integral. In particular, a Cayley graph of a 2-group generated by a normal set of involutions is integral. We prove that a Cayley graph over the symmetric group of degree n no less than 2 generated by all transpositions is integral. We find the spectrum of a Cayley graph over the alternating group of degree n no less than 4 with a generating set of 3-cycles of the form (k i j) with fixed k, as {−n+1, 1−n+1, 22 −n+1, …, (n−1)2 −n+1}.

Extended symmetric spaces and θ-twisted involution graphs

For a Weyl group G and an automorphism θ of order 2, the set of involutions and θ-twisted involutions can be generated by considering actions by basis elements, creating a poset structure on the elements. Haas and Helminck showed that there is a relationship between these sets and their Bruhat posets. We extend that result by considering other bases and automorphisms. We show for G = Sn, θ an involution, and any basis consisting of transpositions, the extended symmetric space is generated by a similar algorithm. Moreover, there is an isomorphism of the poset graphs for certain bases and θ.

Generating sets of involutions of finite simple groups

Generating sets of involutions of finite simple groups

Integral Cayley Graphs

Let $G$ be a group and $S\subseteq G$ its subset such that $S=S^{-1}$, where $S^{-1}=\{s^{-1}\mid s\in S\}$. Then {\it the Cayley graph ${\rm Cay}(G,S)$} is an undirected graph $\Gamma$ with the vertex set $V(\Gamma)=G$ and the edge set $E(\Gamma)=\{(g,gs)\mid g\in G, s\in S\}$. A graph $\Gamma$ is said to be {\it integral} if every eigenvalue of the adjacency matrix of $\Gamma$ is integer. In the paper, we prove the following theorem: {\it if a subset $S=S^{-1}$ of $G$ is normal and $s\in S\Rightarrow s^k\in S$ for every $k\in \mathbb{Z}$ such that $(k,|s|)=1$, then ${\rm Cay}(G,S)$ is integral.} In particular, {\it if $S\subseteq G$ is a normal set of involutions, then ${\rm Cay}(G,S)$ is integral.} We also use the theorem to prove that {\it if $G=A_n$ and $S=\{(12i)^{\pm1}\mid i=3,\dots,n\}$, then ${\rm Cay}(G,S)$ is integral.} Thus, we give positive solutions for both problems 19.50(a) and 19.50(b) in "Kourovka Notebook".

Generating Sets of Involutions of Finite Simple Groups

Generating Sets of Involutions of Finite Simple Groups

Optimal Gray code for involutions

We present a Gray code for the set of involutions of the symmetric group, where the difference between two consecutive involutions is a transposition, or a rotation of three letters. Our code improves previous constructions, and it is proven to be optimal in relation to the Hamming distance between consecutive elements of the code. We also show that the transitions made using a transposition are more frequent than those made using a rotation of three elements.

The Eulerian distribution on involutions is indeed γ-positive

Let In and Jn denote the set of involutions and fixed-point free involutions of {1,…,n}, respectively, and let des(π) denote the number of descents of the permutation π. We prove a conjecture of Guo and Zeng which states that In(t):=∑π∈Intdes(π) is γ-positive for n≥1 and J2n(t):=∑π∈J2ntdes(π) is γ-positive for n≥9. We also prove that the number of (3412,3421)-avoiding permutations with m double descents and k descents is equal to the number of separable permutations with m double descents and k descents.

A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters

We prove Lusztig's conjectures ${\bf P1}$--${\bf P15}$ for the affine Weyl group of type $\tilde{G}_2$ for all choices of parameters. Our approach to compute Lusztig's $\mathbf{a}$-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the $\mathbf{a}$-function and we explicitly construct such a system in type $\tilde{G}_2$ for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type $\tilde{G}_2$, allowing us to prove ${\bf P1}$ and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types $G_2$ and $A_1$, along with some explicit computations for the finite cells.