Characters Of Symmetric Group Research Articles

Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

Positivity of the Symmetric Group Characters Is as Hard as the Polynomial Time Hierarchy

Tame quivers and affine bases I: A Hall algebra approach to the canonical bases

In [17], Lusztig gave an explicit construction of the affine canonical basis by simple perverse sheaves. In this paper, we construct a bar-invariant basis by using a PBW basis arising from representations of the corresponding tame quiver. We prove that this bar-invariant basis coincides with Lusztig's canonical basis and obtains a concrete bijection between the elements in theses two bases. The index set of these bases is listed orderly by modules in preprojective, regular non-homogeneous, preinjective components and irreducible characters of symmetric groups. Our results are based on the work of Lin-Xiao-Zhang ([14]) and closely related with the work of Beck-Nakajima ([2]). A crucial method in our construction is a generalization of that in [8].

Sylow Branching Coefficients and Hook Partitions

We give a description of the irreducible constituents of the restriction to Sylow 2-subgroups of irreducible characters of symmetric groups labelled by hook partitions.

The characters of symmetric groups that depend only on length

The characters of symmetric groups that depend only on length

Normalized characters of symmetric groups and Boolean cumulants via Khovanov's Heisenberg category

In this paper, we study relationships between the normalized characters of symmetric groups and the Boolean cumulants of Young diagrams. Specifically, we show that each normalized character is a polynomial of twisted Boolean cumulants with coefficients being non-negative integers, and conversely, that, when we expand a Boolean cumulant in terms of normalized characters, the coefficients are again non-negative integers. The main tool is Khovanov's Heisenberg category and the recently established connection of its center to the ring of functions on Young diagrams, which enables one to apply graphical manipulations to the computation of functions on Young diagrams. Therefore, this paper is an attempt to deepen the connection between the asymptotic representation theory and graphical categorification.

Non-linear Sylow branching coefficients for symmetric groups

We study the restriction to Sylow subgroups of irreducible characters of symmetric groups. In particular, we give a precise description of the degrees of the irreducible constituents in terms of the shape of the partition that labels a given irreducible character. Our main result is a wide generalization of [Giannelli and Navarro (Proc Am Math Soc 146(5):1963–1976, 2018), Theorem 3.1].

Plethysm and a character embedding problem of Miller

We use a plethystic formula of Littlewood to answer a question of Miller on embeddings of symmetric group characters. We also reprove a result of Miller on character congruences.

Symmetric group characters of almost square shape

We give closed product formulas for the irreducible characters of the symmetric groups related to rectangular `almost square' Young diagrams $p \times(p+\delta)$ for a fixed value of an integer $\delta$ and an arbitrary integer $p$.

Quantum immanants, double Young–Capelli bitableaux and Schur shifted symmetric functions

In this paper, we introduced two classes of elements in the enveloping algebra [Formula: see text]: the double Young–Capelli bitableaux [Formula: see text] and the central Schur elements [Formula: see text], that act in a remarkable way on the highest weight vectors of irreducible Schur modules. Any element [Formula: see text] is the sum of all double Young–Capelli bitableaux [Formula: see text], [Formula: see text] row (strictly) increasing Young tableaux of shape [Formula: see text]. The Schur elements [Formula: see text] are proved to be the preimages — with respect to the Harish-Chandra isomorphism — of the shifted Schur polynomials [Formula: see text]. Hence, the Schur elements are the same as the Okounkov quantum immanants, recently described by the present authors as linear combinations of Capelli immanants. This new presentation of Schur elements/quantum immanants does not involve the irreducible characters of symmetric groups. The Capelli elements [Formula: see text] are column Schur elements and the Nazarov elements [Formula: see text] are row Schur elements. The duality in [Formula: see text] follows from a combinatorial description of the eigenvalues of the [Formula: see text] on irreducible modules that is dual (in the sense of shapes/partitions) to the combinatorial description of the eigenvalues of the [Formula: see text]. The passage [Formula: see text] for the algebras [Formula: see text] is obtained both as direct and inverse limit in the category of filtered algebras, via the Olshanski decomposition/projection.

Covering numbers for characters of symmetric groups

Covering numbers for characters of symmetric groups

Word Measures on Symmetric Groups

Abstract Fix a word $ w $ in a free group $ \textbf {F}$ on $r$ generators. A $w$-random permutation in the symmetric group $S_{N}$ is obtained by sampling $r$ independent uniformly random permutations $ \sigma _{1},\ldots ,\sigma _{r}\in S_{N}$ and evaluating $w\left (\sigma _{1},\ldots ,\sigma _{r}\right )$. In [39, 40], it was shown that the average number of fixed points in a $w$-random permutation is $1+\theta \left (N^{1-\pi \left (w\right )}\right )$, where $ \pi \left (w\right )$ is the smallest rank of a subgroup $H\le \textbf {F}$ containing $w$ as a non-primitive element. We show that $ \pi \left (w\right )$ plays a role in estimates of all stable characters of symmetric groups. In particular, we show that for all $t\ge 2$, the average number of $t$-cycles is $ \frac {1}{t}+O\left (N^{-\pi \left (w\right )}\right )$. As an application, we prove that for every $s$, every $ \varepsilon>0$ and every large enough $r$, Schreier graphs with $r$ random generators depicting the action of $S_{N}$ on $s$-tuples, have 2nd eigenvalue at most $2\sqrt {2r-1}+\varepsilon $ asymptotically almost surely. An important ingredient in this work is a systematic study of not necessarily connected Stallings core graphs.

Symmetric group characters as symmetric functions

We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of GLn modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.

On quasi Steinberg characters of symmetric and alternating groups and their double covers

An irreducible character of a finite group [Formula: see text] is called quasi [Formula: see text]-Steinberg character for a prime [Formula: see text] if it takes a nonzero value on every [Formula: see text]-regular element of [Formula: see text]. In this paper, we classify the quasi [Formula: see text]-Steinberg characters of Symmetric ([Formula: see text]) and Alternating ([Formula: see text]) groups and their double covers. In particular, an existence of a nonlinear quasi [Formula: see text]-Steinberg character of [Formula: see text] implies [Formula: see text] and of [Formula: see text] implies [Formula: see text].

The Hopf structure of symmetric group characters as symmetric functions

In [24] the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.

Young–Capelli bitableaux, Capelli immanants in U(gl(n)) and the Okounkov quantum immanants

The set of standard Capelli bitableaux and the set of standard Young–Capelli bitableaux are bases of [Formula: see text], whose action on the Gordan–Capelli basis of polynomial algebra [Formula: see text] have remarkable properties (see, e.g. [A. Brini, A. Palareti and A. Teolis, Gordan–Capelli series in superalgebras, Proc. Natl. Acad. Sci. USA 85 (1988) 1330–1333; A. Brini and A. Teolis, Young–Capelli symmetrizers in superalgebras, Proc. Natl. Acad. Sci. USA 86 (1989) 775–778; A. Brini and A. Teolis, Capelli bitableaux and [Formula: see text]-forms of general linear Lie superalgebras, Proc. Natl. Acad. Sci. USA 87 (1990) 56–60; A. Brini and A. Teolis, Capelli’s theory, Koszul maps, and superalgebras, Proc. Natl. Acad. Sci. USA 90 (1993) 10245–10249.]). We introduce a new class of elements of [Formula: see text], called the Capelli immanants, that can be efficiently computed and provide a system of linear generators of [Formula: see text]. The Okounkov quantum immanants [A. Okounkov, Quantum immanants and higher Capelli identities, Transform Groups 1 (1996) 99–126; A. Okounkov, Young basis, Wick formula, and higher Capelli identities, Int. Math. Res. Not. 1996(17) (1996) 817–839.] — quantum immanants, for short — are proved to be simple linear combinations of diagonal Capelli immanants, with explicit coefficients. Quantum immanants can also be expressed as sums of double Young–Capelli bitableaux. Since double Young–Capelli bitableaux uniquely expands into linear combinations of standard Young–Capelli bitableaux, this leads to canonical presentations of quantum immanants, and, furthermore, it does not involve the computation of the irreducible characters of symmetric groups.

Set-partition tableaux and representations of diagram algebras

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation.

Products of symmetric group characters

In [33], the authors introduced a new basis of the ring of symmetric functions which evaluate to the irreducible characters of the symmetric group at roots of unity. The structure coefficients for this new basis are the stable Kronecker coefficients. In this paper we give combinatorial descriptions for several products. In addition, we identify some applications and instances where special cases of these products have occurred elsewhere in the mathematical literature.

Hook removal operators on the odd Young graph

In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.

Liminal reciprocity and factorization statistics

Let M d,n (q) denote the number of monic irreducible polynomials in 𝔽 q [x 1 ,x 2 ,...,x n ] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of a result of Church, Ellenberg, and Farb.

On parity and characters of symmetric groups

We present a conjectural parity bias in the character values of the symmetric group. The main conjecture says that a character value chosen uniformly at random from the character table of Sn is congruent to 0mod2 with probability →1 as n→∞. A more general conjecture says that the same is true for all primes p, not only p=2. We relate these conjectures to zeros, give generating functions for computing lower bounds, and present some computational data in support of the main conjecture.