Representations Of Symmetric Group Research Articles

Generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic 2

AbstractWe prove generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic 2, similar to Donkin’s results for decomposition numbers of representations of symmetric groups.

Existence of elementwise invariant vectors in representations of symmetric groups

Existence of elementwise invariant vectors in representations of symmetric groups

Decomposition numbers of 2-parts spin representations of symmetric groups in characteristic 2

We give explicit formulas to compute most of the decomposition numbers of reductions modulo 2 of irreducible spin representations of symmetric groups indexed by partitions with at most 2 parts. In many of the still open cases small upper bounds are found.

Kronecker comultiplication of stable characters and restriction from Smn to Sm × Sn

A family of symmetric functions s˜λ was introduced in [11], and independently in [1]. The s˜λ encode many stability properties of representations of symmetric groups (e.g. when multiplied, the structure constants are reduced Kronecker coefficients). We show that the structure constants for the Kronecker comultiplication Δ⁎ are multiplicities for the restriction of irreducible representations from Smn to Sm×Sn (provided m and n are sufficiently large), and use the structure of s˜λ to demonstrate two-row stability properties of these restriction multiplicities.

A q-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets

Let f(z)=∑ n=0 ∞ (-1) n z n /n!n!. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1/f(z)=∑ n=0 ∞ ω n z n /n!n!. They proved that ω n counts the number of pairs of permutations of the nth symmetric group 𝒮 n with no common ascent. This paper gives a combinatorial interpretation of a natural q-analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n (q) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: ∑ i=0 n (-1) i e i h n-i =0, which then generalizes our q-analogue to a symmetric group representation result.

Indecomposable objects in Khovanov–Sazdanovic's generalizations of Deligne's interpolation categories

Khovanov and Sazdanovic recently introduced symmetric monoidal categories parameterized by rational functions and given by quotients of categories of two-dimensional cobordisms. These categoriesgeneralizeDeligne'sinterpolation categories of representations of symmetric groups. In this paper, we classify indecomposable objects and identify the associated graded Grothendieck rings of Khovanov–Sazdanovic's categories through sums of representation categories over crossed products of polynomial rings over a general field. To obtain these results, we introduce associated graded categories for Krull–Schmidt categories with filtrations as a categorification of the associated graded Grothendieck ring, and study field extensions and Galois descent for Krull–Schmidt categories.

Categorifications of rational Hilbert series and characters of FSop modules

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid.

A web basis of invariant polynomials from noncrossing partitions

The irreducible representations of symmetric groups can be realized as certain graded pieces of invariant rings, equivalently as global sections of line bundles on partial flag varieties. There are various ways to choose useful bases of such Specht modules Sλ. Particularly powerful are web bases, which make important connections with cluster algebras and quantum link invariants. Unfortunately, web bases are only known in very special cases—essentially, only the cases λ=(d,d) and λ=(d,d,d). Building on work of B. Rhoades (2017), we construct an apparent web basis of invariant polynomials for the 2-parameter family of Specht modules with λ of the form (d,d,1ℓ). The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier enumerative results in combinatorial dynamics.

Defect 2 spin blocks of symmetric groups and canonical basis coefficients

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect 2 2 , analogous to Richards’s formula for defect 2 2 blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding “ q q -decomposition numbers”, i.e. the canonical basis coefficients in the level- 1 1 q q -deformed Fock space of type A 2 n ( 2 ) A^{(2)}_{2n} ; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic 2 n + 1 2n+1 . Along the way, we prove some general results on q q -decomposition numbers. This paper represents the first substantial progress on canonical bases in type A 2 n ( 2 ) A^{(2)}_{2n} .

Stability phenomena for resonance arrangements

We prove that the i th i^\text {th} graded pieces of the Orlik–Solomon algebras or Cordovil algebras of resonance arrangements form a finitely generated FS o p \operatorname {FS}^{\mathrm {op}} -module, thus obtaining information about the growth of their dimensions and restrictions on the irreducible representations of symmetric groups that they contain.

Lifting images of standard representations of symmetric groups

We investigate closed subgroups \(G \subseteq \mathrm {Sp}_{2g}(\mathbb {Z}_2)\) whose modulo-2 images coincide with the image \(\mathfrak {S}_{2g + 1} \subseteq \mathrm {Sp}_{2g}(\mathbb {F}_2)\) of \(S_{2g + 1}\) or the image \(\mathfrak {S}_{2g + 2} \subseteq \mathrm {Sp}_{2g}(\mathbb {F}_2)\) of \(S_{2g + 2}\) under the standard representation. We show that when \(g \ge 2\), the only closed subgroup \(G \subseteq \mathrm {Sp}_{2g}(\mathbb {Z}_2)\) surjecting onto \(\mathfrak {S}_{2g + 2}\) is its full inverse image in \(\mathrm {Sp}_{2g}(\mathbb {Z}_2)\), while all subgroups \(G \subseteq \mathrm {Sp}_{2g}(\mathbb {Z}_2)\) surjecting onto \(\mathfrak {S}_{2g + 1}\) are open and contain the level-8 principal congruence subgroup of \(\mathrm {Sp}_{2g}(\mathbb {Z}_2)\). As an immediate application, we are able to strengthen a result of Zarhin on 2-adic Galois representations associated to hyperelliptic curves. We also prove an elementary corollary concerning even-degree polynomials with full Galois group.

Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory

The Kronecker, plethysm and Sylow branching coefficients describe the decomposition of representations of symmetric groups obtained by tensor products and induction. Understanding these decompositions has been hailed as one of the definitive open problems in algebraic combinatorics and has profound and deep connections with representation theory, symplectic geometry, complexity theory, quantum information theory, and local-global conjectures in representation theory of finite groups. The overarching theme of the Mini-Workshop has been the use of hidden, richer representation theoretic structures to prove and disprove conjectures concerning these coefficients. These structures arise from the modular and local-global representation theory of symmetric groups, graded representation theory of Hecke and Cherednik algebras, and categorical Lie theory.

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

AbstractWebs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

Composition factors of 2-parts spin representations of symmetric groups

Given an odd prime $p$, we identify composition factors of the reduction modulo $p$ of spin irreducible representations of the covering groups of symmetric groups indexed by partitions with 2 parts and find some decomposition numbers.

Generalized Springer Correspondence for Symmetric Spaces Associated to Orthogonal Groups

Let $G = GL_N(\mathbf{k})$, where $\mathbf{k}$ is an algebraically closed field of $\operatorname{ch} \mathbf{k} \ne 2$, and $\theta$ an involutive automorphism of $G$ such that $H = (G^{\theta})^0$ is isomorphic to $SO_N(\mathbf{k})$. Then $G^{\iota\theta} = \{ g \in G \mid \theta(g) = g^{-1}\}$ is regarded as a symmetric space $G/G^{\theta}$. Let $G^{\iota\theta}_{\operatorname{uni}}$ be the set of unipotent elements in $G^{\iota\theta}$. $H$ acts on $G^{\iota\theta}_{\operatorname{uni}}$ by the conjugation. As an analogue of the generalized Springer correspondence in the case of reductive groups, we establish in this paper the generalized Springer correspondence between $H$-orbits in $G^{\iota\theta}_{\operatorname{uni}}$ and irreducible representations of various 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.

Lower bounds for dimensions of irreducible representations of symmetric groups

We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.

The Heisenberg product seen as a branching problem for connected reductive groups, stability properties

Abstract In this article, we study, in the context of complex representations of symmetric groups, some aspects of the Heisenberg product, introduced by Marcelo Aguiar, Walter Ferrer Santos and Walter Moreira in 2017. When applied to irreducible representations, this product gives rise to the Aguiar coefficients. We prove that these coefficients are in fact also branching coefficients for representations of connected complex reductive groups. This allows to use geometric methods already developed in a previous article, notably based on notions from geometric invariant theory, and to obtain some stability results on Aguiar coefficients, generalising some of the results concerning them given by Li Ying.

Spinorial representations of symmetric groups

A real representation π of a finite group may be regarded as a homomorphism to an orthogonal group O(V). For symmetric groups Sn, alternating groups An, and products Sn×Sn′ of symmetric groups, we give criteria for whether π lifts to the double cover Pin(V) of O(V), in terms of character values. From these criteria we compute the second Stiefel-Whitney classes of these representations.

Horn inequalities for nonzero Kronecker coefficients

The Kronecker coefficients gαβγ and the Littlewood- Richardson coefficients cαβγ are nonnegative integers depending on three partitions α, β, and γ. By definition, gαβγ (resp. cαβγ) are the multiplicities of the tensor product decomposition of two irreducible representations of symmetric groups (resp. linear groups). By a classical Littlewood-Murnaghan's result the Kronecker coefficients extend the Littlewood-Richardson ones.The nonvanishing of the Littlewood-Richardson coefficient cαβγ implies that (α,β,γ) satisfies some linear inequalities called Horn inequalities. In this paper, we extend the essential Horn inequalities to the triples of partitions corresponding to a nonzero Kronecker coefficient.Along the way, we describe the set of tripless (α,β,γ) of partitions such that cαβγ≠0 and l(α)≤e, l(β)≤f and l(γ)≤e+f, for some given positive integers e and f. This set is the natural analogue of the classical Horn semigroup when one thinks about cαβγ as the branching multiplicities for the subgroup GLe×GLf of GLe+f.