Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group, i.e. that conjugacy classes of S_n do not unite under the natural representation. We prove this fact, and give its application to the enumeration of fixed points under a natural action of S_n x S_n. We also consider the permutation representations of S_n which arise from the action of S_n on k-tuples, and classify which of them unite conjugacy classes and which do not.
We prove a convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool we introduce new semigroup of partial permutations. We describe its structure, representations, and characters. We also discuss filtrations on the subalgebra of invariants in the semigroup algebra.
A now-classical cyclic extension of the descent set of a permutation has been introduced by Klyachko and Cellini. Following a recent axiomatic approach to this notion, it is natural to ask which sets of permutations admit such an extension. The main result of this paper is a complete answer in the case of conjucay classes of permutations. It is shown that the conjugacy class of cycle type $λ$ has such an extension if and only if $λ$ is not of the form $(r^s)$ for some square-free $r$. The proof involves a detailed study of hook constituents in higher Lie characters.
The number of peaks of a random permutation is known to be asymptotically normal. We give a new proof of this and prove a central limit theorem for the distribution of peaks in a fixed conjugacy class of the symmetric group. Our technique is to apply ``analytic combinatorics'' to study a complicated but exact generating function for peaks in a given conjugacy class.
A conjugacy class $C$ of a finite group $G$ is a sign conjugacy class if every irreducible character of $G$ takes value 0, 1 or -1 on $C$. In this paper we classify the sign conjugacy classes of the symmetric groups and thereby verify a conjecture of Olsson.
For a prime $p$, we say that a conjugacy class of a finite group $G$ is $p$-vanishing if every irreducible character of $G$ of degree divisible by $p$ takes value 0 on that conjugacy class. In this paper we completely classify 2-vanishing and 3-vanishing conjugacy classes for the symmetric group and do some work in the classification of $p$-vanishing conjugacy classes of the symmetric group for $p\geq 5$. This answers a question by Navarro for $p=2$ and $p=3$ and partly answers it for $p\geq 5$.
For a fixed permutation $σ\in S_k$, let $N_σ$ denote the function which counts occurrences of $σ$ as a pattern in permutations from $S_n$. We study the expected value (and $d$-th moments) of $N_σ$ on conjugacy classes of $S_n$ and prove that the irreducible character support of these class functions stabilizes as $n$ grows. This says that there is a single polynomial in the variables $n, m_1, \ldots, m_{dk}$ which computes these moments on any conjugacy class (of cycle type $1^{m_1}2^{m_2}\cdots$) of any symmetric group. This result generalizes results of Hultman and of Gill, who proved the ca…
We prove a conjecture raised by the work of Diaconis and Shahshahani (1981) about the mixing time of random walks on the permutation group induced by a given conjugacy class. To do this we exploit a connection with coalescence and fragmentation processes and control the Kantorovitch distance by using a variant of a coupling due to Oded Schramm. Recasting our proof in the language of Ricci curvature, our proof establishes the occurrence of a phase transition, which takes the following form in the case of random transpositions: at time $cn/2$, the curvature is asymptotically zero for $c\le 1$ an…
We discuss permutation representations which are obtained by the natural action of $S_n \times S_n$ on some special sets of invertible matrices, defined by simple combinatorial attributes. We decompose these representations into irreducibles. The multiplicities involved have a nice combinatorial interpretation. We also generalize known results on asymptotic behavior of the conjugacy representation of $S_n$.
The distribution of descents in fixed conjugacy classes of $S_n$ has been studied, and it is shown that its moments have interesting properties. Fulman proved that the descent numbers of permutations in conjugacy classes with large cycles are asymptotically normal, and Kim proved that the descent numbers of fixed point free involutions are also asymptotically normal. In this paper, we generalize these results to prove a central limit theorem for descent numbers of permutations in any conjugacy class of $S_n$.
Given a permutation statistic $s : S_n \to \mathbb{R}$, define the mean statistic $\bar{s}$ as the statistic which computes the mean of $s$ over conjugacy classes. We describe a way to calculate the expected value of $s$ on a product of $t$ independently chosen elements from the uniform distribution on a union of conjugacy classes $Γ\subseteq S_n$. In order to apply the formula, one needs to express the class function $\bar{s}$ as a linear combination of irreducible $S_n$-characters. We provide such expressions for several commonly studied permutation statistics, including the excedance number…
The $d$-Simultaneous Conjugacy problem in the symmetric group $S_n$ asks whether there exists a permutation $τ\in S_n$ such that $b_j = τ^{-1}a_j τ$ holds for all $j = 1,2,\ldots, d$, where $a_1, a_2,\ldots , a_d$ and $b_1, b_2,\ldots , b_d$ are given sequences of permutations in $S_n$. The time complexity of existing algorithms for solving the problem is $O(dn^2)$. We show that for a given positive integer $d$ the $d$-Simultaneous Conjugacy problem in $S_n$ can be solved in $o(n^2)$ time.
Given a permutation pattern p and an equivalence relation on permutations, we study the corresponding equivalence classes all of whose members avoid p. Four relations are studied: Conjugacy, order isomorphism, Knuth-equivalence and toric equivalence. Each of these produces a known class of permutations or a known counting sequence. For example, involutions correspond to conjugacy, and permutations whose insertion tableau is hook-shaped with 2 in the first row correspond to Knuth-equivalence. These permutations are equinumerous with certain congruence classes of graph endomorphisms. In the case…
By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type $A$ are found. This leads to new models of card shuffling and has interesting combinatorial and numb…
We construct an infinite word $w$ over the $5$-letter alphabet such that for every factor $f$ of $w$ of length at least two, there exists a cyclic permutation of $f$ that is not a factor of $w$. In other words, $w$ does not contain a non-trivial conjugacy class. This proves the conjecture in Gamard et al. [TCS 2018]
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily: to each conjugacy class we associate a two-dimensional surface and the asymptotic properties of the conjugacy class depend only on the genus of the resulting surface. This construction closely resembles …
The transitive simultaneous conjugacy problem asks whether there exists a permutation $τ\in S_n$ such that $b_j = τ^{-1} a_j τ$ holds for all $j = 1,2, \ldots, d$, where $a_1, a_2, \ldots, a_d$ and $b_1, b_2, \ldots, b_d$ are given sequences of $d$ permutations in $S_n$, each of which generates a transitive subgroup of $S_n$. As from mid 70' it has been known that the problem can be solved in $O(dn^2)$ time. An algorithm with running time $O(dn \log(dn))$, proposed in late 80', does not work correctly on all input data. In this paper we solve the transitive simultaneous conjugacy problem in $O…
The convolution of indicators of two conjugacy classes on the symmetric group S_q is usually a complicated linear combination of indicators of many conjugacy classes. Similarly, a product of the moments of the Jucys--Murphy element involves many conjugacy classes with complicated coefficients. In this article we consider a combinatorial setup which allows us to manipulate such products easily and we show that it very closely related to the combinatorial approach to random matrices. Our formulas are exact (in a sense that they hold not only asymptotically for large q). This result has many inte…
This is a note for constructing fundamental invariants and computing the Hilbert series of the invariant subalgebras of tensor products of polynomial rings under the action by a direct product of symmetric groups. Our computation relies on Schur functions bringing together several identities of combinatorial generating functions including that of plane partitions.
For a Coxeter group (W,S), a permutation of the set S is called a Coxeter word and the group element represented by the product is called a Coxeter element. Moving the first letter to the end of the word is called a rotation and two Coxeter elements are rotation equivalent if their words can be transformed into each other through a sequence of rotations and legal commutations. We prove that Coxeter elements are conjugate if and only if they are rotation equivalent. This was known for some special cases but not for Coxeter groups in general.
We introduce a notion of {\em cyclic Schur-positivity} for sets of permutations, which naturally extends the classical notion of Schur-positivity, and it involves the existence of a bijection from permutations to standard Young tableaux that preserves the cyclic descent set. Cyclic Schur-positive sets of permutations are always Schur-positive, but the converse does not hold, as exemplified by inverse descent classes, Knuth classes and conjugacy classes. In this paper we show that certain classes of permutations invariant under either horizontal or vertical rotation are cyclic Schur-positive. T…
In this paper, we define a set which has a finite group action and is generated by a finite color set, a set which has a finite group action, and a subset of the set of non negative integers. we state its properties to apply one of solution of the following two problems, respectively. First, we calculate the generating function of the character of symmetric powers of permutation representation associated with a set which has a finite group action. Second, we calculate the number of primitive colorings on some objects of polyhedrons. It is a generalization of the calculation of the number of pr…
We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite simple groups, random walks on Chevalley groups, the final solution to the Ore conjecture about commutators in finite simple groups and other similar problems. In this paper, we solve a strong version of the Boston-Shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups (t…
We call an irreducible character $p$-singular if $p$ divides its degree. We prove a number of equivalent conditions for a character of the symmetric group $S_n$ to be $p$-singular, involving a certain family of conjugacy classes. This answers in part a question by Navarro.
Two distinct projections of finite rank $m$ are adjacent if their difference is an operator of rank two or, equivalently, the intersection of their images is $(m-1)$-dimensional. We extend this adjacency relation on other conjugacy classes of finite-rank self-adjoint operators which leads to a natural generalization of Grassmann graphs. Let ${\mathcal C}$ be a conjugacy class formed by finite-rank self-adjoint operators with eigenspaces of dimension greater than $1$. Under the assumption that operators from ${\mathcal C}$ have at least three eigenvalues we prove that every automorphism of the …
This note contains two results on the distribution of crossing numbers and nesting numbers in permutations of type B. More precisely, we prove a Bn- analogue of the symmetric distribution of crossings and nestings of permutations due to Corteel (Adv. in Appl. Math., 38(2)(2007), 149-163) as well as the sym- metric distribution of k-crossings and k-nestings of permutations due to Burrill et al. (DMTCS proc. AN, (2010), 461-468).
Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify a subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new par…
Generalizing the notion of a vexillary permutation, we introduce a filtration of S_infinity by the number of Schur function terms in the Stanley symmetric function, with the kth filtration level called the k-vexillary permutations. We show that for each k, the k-vexillary permutations are characterized by avoiding a finite set of patterns. A key step is the construction of a Specht series, in the sense of James and Peel, for the Specht module associated to the diagram of a permutation. As a corollary, we prove a conjecture of Liu on diagram varieties for certain classes of permutation diagrams…
We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's cyclic valley-hopping action as well as a formula of Brenti. Along the way, we give a new proof for the $γ$-positivity of the excedance number distribution over any fixed conjugacy class along with a combinatorial interpretation of the $γ$-coefficients.
We consider random permutations which are spherically symmetric with respect to a metric on the symmetric group $S_n$ and are consistent as $n$ varies. The extreme infinitely spherically symmetric permutation-valued processes are identified for the Hamming, Kendall-tau and Caley metrics. The proofs in all three cases are based on a unified approach through stochastic monotonicity.
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $σ$ is the order type of the restriction of $σ: [n] \to [n]$ to a subset $S \subset [n]$. First, is it possible for the pattern counts in a permutation to be exactly equal to their expected values under a uniform distribution? Attempts to address this question lead naturally to an interesting number theoretic problem: when does $k!$ divide $\binom{n}{k}$? Second, if the tensor product of a permutation with large random permutations is random-lik…
We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric functions. After a suitable specialization the new identities reduce to identities involving the first and second order Eulerian polynomials. The study of these identities led us to consider a family of symmetric functions associated with a class of permutations introduced by Gessel and Stanley, known in the literature as Stirling permutations. In particular, we define certain type statistics on Stirling permutations that refine the s…
We prove conjectures of the third author [L. Tevlin, Proc. FPSAC'07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers.
We present (bi-)symmetric generating functions for the joint distributions of Euler-Stirling statistics on permutations, including the number of descents ($\mathsf{des}$), inverse descents ($\mathsf{ides}$), the number of left-to-right maxima ($\mathsf{lmax}$), the number of right-to-left maxima ($\mathsf{rmax}$) and the number of left-to-right minima ($\mathsf{lmin}$). We also show how they recover the classical symmetric generating function of permutations due to Carlitz, Roselle and Scoville (1966). Our proofs exploit three different recursive constructions of inversion sequences, bijection…
We consider the graph whose vertex set is a conjugacy class ${\mathcal C}$ consisting of finite-rank self-adjoint operators on a complex Hilbert space $H$. The dimension of $H$ is assumed to be not less than $3$. In the case when operators from ${\mathcal C}$ have two eigenvalues only, we obtain the Grassmann graph formed by $k$-dimensional subspaces of $H$, where $k$ is the smallest dimension of eigenspaces. Classical Chow's theorem describes automorphisms of this graph for $k>1$. Under the assumption that operators from ${\mathcal C}$ have more than two eigenvalues we show that every automor…
We generalize the work of Fomin, Greene, Reiner, and Shimozono on balanced labellings in two directions: (1) we define the diagrams of affine permutations and the balanced labellings on them; (2) we define the set-valued version of the balanced labellings. We show that the column-strict balanced labellings on the diagram of an affine permutation yield the affine Stanley symmetric function defined by Lam, and that the column-strict set-valued balanced labellings yield the affine stable Grothendieck polynomial of Lam. Moreover, once we impose suitable flag conditions, the flagged column-strict s…
The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant $γ_G(M)$ for a finite dimensional module $M$ of a finite group $G$ which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for all the signed permutation modules of the symmetric group using tools from representation theory and combinatorics.
In [Boltje,Hartmann: Permutation resolutions for Specht modules, J. Algebraic Combin. 34 (2011), 141-162], a chain complex was constructed in a combinatorial way which conjecturally is a resolution of the (dual of the) integral Specht module for the symmetric group in terms of permutation modules. In this paper we extend the definition of the chain complex to the integral Iwahori Hecke algebra and prove the same partial exactness results that were proved in the symmetric group case. A complete proof of the exactness conjecture in the symmetric group case was recently given by Santana and Yudin…
We use the theory of symmetric functions to enumerate various classes of alternating permutations w of {1,2,...,n}. These classes include the following: (1) both w and w^{-1} are alternating, (2) w has certain special shapes, such as (m-1,m-2,...,1), under the RSK algorithm, (3) w has a specified cycle type, and (4) w has a specified number of fixed points. We also enumerate alternating permutations of a multiset. Most of our formulas are umbral expressions where after expanding the expression in powers of a variable E, E^k is interpreted as the Euler number E_k. As a small corollary, we obtai…
This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from…
The following long-standing problem in combinatorics was first posed in 1993 by Gessel and Reutenauer. For which multisubsets $B$ of the symmetric group $\fS_n$ is the quasisymmetric function $$Q(B) = \sum_{π\in B}F_{\Des(π), n}$$ a symmetric function? Here $\Des(π)$ is the descent set of $π$ and $F_{\Des(π), n}$ is Gessel's fundamental basis for the vector space of quasisymmetric functions. The purpose of this paper is to provide a useful characterization of these multisets. Using this characterization we prove a conjecture of Elizalde and Roichman. Two other corollaries are also given. The f…
We consider several generalizations of the classical $γ$-positivity of Eulerian polynomials (and their derangement analogues) using generating functions and combinatorial theory of continued fractions. For the symmetric group, we prove an expansion formula for inversions and excedances as well as a similar expansion for derangements. We also prove the $γ$-positivity for Eulerian polynomials for derangements of type $B$. More general expansion formulae are also given for Eulerian polynomials for $r$-colored derangements. Our results answer and generalize several recent open problems in the lite…
Descents in permutations or words are defined from the relative position of two consecutive letters. We investigate a statistic involving patterns of k consecutive letters, and show that it leads to Hopf algebras generalizing noncommutative symmetric functions and quasi-symmetric functions.
