Schur Modules Research Articles

A representation-theoretic interpretation of positroid classes

A positroid is the matroid of a real matrix with nonnegative maximal minors, a positroid variety is the closure of the locus of points in a complex Grassmannian whose matroid is a fixed positroid, and a positroid class is the cohomology class Poincaré dual to a positroid variety. We define a family of representations of general linear groups whose characters are symmetric polynomials representing positroid classes. These representations are certain diagram Schur modules in the sense of James and Peel. This gives a new algebraic interpretation of the Schubert structure constants for the product of a Schubert polynomial and Schur polynomial, and of the 3-point Gromov-Witten invariants for Grassmannians, proving a conjecture of Postnikov. As a byproduct, we obtain an effective algorithm for decomposing positroid classes into Schubert classes.

Kohnert's rule for flagged Schur modules

Flagged Schur modules generalize the irreducible representations of the general linear group under the action of the Borel subalgebra. Their characters include many important generalizations of Schur polynomials, such as Demazure characters, flagged skew Schur polynomials, and Schubert polynomials. In this paper, we prove the characters of flagged Schur modules can be computed using a simple combinatorial algorithm due to Kohnert if and only if the indexing diagram is northwest. This gives a new proof that characters of flagged Schur modules are nonnegative sums of Demazure characters and gives a representation theoretic interpretation for Kohnert polynomials.

Polarizations and hook partitions

In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies–Welker to show that the L-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the “spanning tree condition” of Almousa–Fløystad–Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called “restricted powers” of the graded maximal ideal.

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.

Multiplicity-free skew Schur polynomials

We provide a non-recursive, combinatorial classification of multiplicity-free skew Schur polynomials. These polynomials are $GL_n$, and $SL_n$, characters of the skew Schur modules. Our result extends work of H. Thomas--A. Yong, and C. Gutschwager, in which they classify the multiplicity-free skew Schur functions.

A Demazure Character Formula for the Product Monomial Crystal

The product monomial crystal was defined by Kamnitzer, Tingley, Webster, Weekes, and Yacobi for any semisimple simply-laced Lie algebra $\mathfrak{g}$, and depends on a collection of parameters $\mathbf{R}$. We show that a family of truncations of this crystal are Demazure crystals, and give a Demazure-type formula for the character of each truncation, and the crystal itself. This character formula shows that the product monomial crystal is the crystal of a generalised Demazure module, as defined by Lakshmibai, Littelmann and Magyar. In type $A$, we show the product monomial crystal is the crystal of a generalised Schur module associated to a column-convex diagram depending on $\mathbf{R}$.

Specht modules decompose as alternating sums of restrictions of Schur modules

Schur modules give the irreducible polynomial representations of the general linear group G L t \mathrm {GL}_t . Viewing the symmetric group S t \mathfrak {S}_t as a subgroup of G L t \mathrm {GL}_t , we may restrict Schur modules to S t \mathfrak {S}_t and decompose the result into a direct sum of Specht modules, the irreducible representations of S t \mathfrak {S}_t . We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for S t \mathfrak {S}_t for t t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.

Acyclicity of Schur complexes and torsion freeness of Schur modules

Let R be a Noetherian commutative ring and M a R-module with pdRM≤1 that has rank. Necessary and sufficient conditions were provided in [1] for an exterior power ∧kM to be torsion free. When M is an ideal of R similar necessary and sufficient conditions were provided in [2] for a symmetric power SkM to be torsion free. We extend these results to a broad class of Schur modules Lλ/μM. En route, for any map of finite free R modules ϕ:F→G we also study the general structure of the Schur complexes Lλ/μϕ, and provide necessary and sufficient conditions for the acyclicity of any given Lλ/μϕ by computing explicitly the radicals of the ideals of maximal minors of all its differentials.

Injective Schur modules

We determine the partitions λ for which the corresponding induced module (or Schur module in the language of Buchsbaum et al., [1]) ∇(λ) is injective in the category of polynomial modules for a general linear group over an infinite field, equivalently which Weyl modules are projective polynomial modules. Since the problem is essentially no more difficult in the quantised case we address it at this level of generality. Expressing our results in terms of the representation theory of Hecke algebras at the parameter q we determine the partitions λ for which the corresponding Specht module is a Young module, when 1+q≠0. In the classical case this problem was addressed by D. Hemmer, [10]. The nature of the set of partitions appearing in our solution gives a new formulation of Carter's condition on regular partitions. On the other hand, we note, in Remark 2.22, that the result on irreducible Weyl modules for the quantised Schur algebra Sq(n,n), [15], Theorem 5.39, given in terms of Carter partitions, may be also used to obtain the main result presented here.

Test, multiplier and invariant ideals

This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V)×GL(W)-invariant ideal in S=Sym(V⊗W⁎), where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal.Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we compute the test ideals (and so the F-pure thresholds) of any sum of products of determinantal ideals. Not all the invariant ideals are as the latter (not even in characteristic 0), but they are up to integral closure, and this is enough to reach our goals.The results concerning the test ideals are obtained as a consequence of general results holding true in a special situation. Within such framework fall determinantal objects of a generic matrix, as well as of a symmetric matrix and of a skew-symmetric one. Similar results are thus deduced for the GL(V)-invariant ideals in Sym(Sym2V) and in Sym(⋀2V). (Monomial ideals also fall in this framework, thus we recover Howald's formula for their multiplier ideals and, more generally, Hara–Yoshida's formula for their test ideals.) In the proof, we introduce the notion of “floating test ideals”, a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities. As will be shown, products of determinantal ideals, and by passing to characteristic 0 ideals generated by a single Schur module, have this property.

PARTITIONS OF SINGLE EXTERIOR TYPE

We characterize the irreducible representations of the general linear group GL(V) that have multiplicity 1 in the direct sum of all Schur modules of a given exterior power of V. These have come up in connection with the relations of the lower order minors of a generic matrix. We show that the minimal relations conjectured by Bruns, Conca and Varbaro are exactly those coming from partitions of single exterior type.

Homotopy transfer and self-dual Schur modules

We consider the free 2-nilpotent graded Lie algebra $$\mathfrak{g}$$ generated in degree one by a finite dimensional vector space V. We recall the beautiful result that the cohomology $$H^ \cdot \left( {\mathfrak{g},\mathbb{K}} \right)$$ of $$\mathfrak{g}$$ with trivial coefficients carries a GL(V)-representation having only the Schur modules V with self-dual Young diagrams {λ: λ = λ′} in its decomposition into GL(V)-irreducibles (each with multiplicity one). The homotopy transfer theorem due to Tornike Kadeishvili allows to equip the cohomology of the Lie algebra g with a structure of homotopy commutative algebra.

POSITIVITY FOR REGULAR CLUSTER CHARACTERS IN ACYCLIC CLUSTER ALGEBRAS

Let Q be an acyclic quiver and let [Formula: see text] be the corresponding cluster algebra. Let H be the path algebra of Q over an algebraically closed field and let M be an indecomposable regular H-module. We prove the positivity of the cluster characters associated to M expressed in the initial seed of [Formula: see text] when either H is tame and M is any regular H-module, or H is wild and M is a regular Schur module which is not quasi-simple.

Secant varieties of ℙ2 × ℙ n embedded by 𝒪(1, 2)

We describe the defining ideal of the rth secant variety of P^2 x P^n embedded by O(1,2), for arbitrary n and r at most 5. We also present the Schur module decomposition of the space of generators of each such ideal. Our main results are based on a more general construction for producing explicit matrix equations that vanish on secant varieties of products of projective spaces. This extends previous work of Strassen and Ottaviani.

Matching polytopes and Specht modules

We prove that the dimension of the Specht module of a forest $G$ is the same as the normalized volume of the matching polytope of $G$. We also associate to $G$ a symmetric function $s_G$ (analogous to the Schur symmetric function $s_\lambda$ for a partition $\lambda$) and investigate its combinatorial and representation-theoretic properties in relation to the Specht module and Schur module of $G$. We then use this to define notions of standard and semistandard tableaux for forests.

Unfolding mixed-symmetry fields in AdS and the BMV conjecture: II. Oscillator realization

Following the general formalism presented in arXiv:0812.3615 — referred to as Paper I — we derive the unfolded equations of motion for tensor fields of arbitrary shape and mass in constantly curved backgrounds by radial reduction of Skvortsov's equations in one higher dimension. The complete unfolded system is embedded into a single master field, valued in a tensorial Schur module realized equivalently via either bosonic (symmetric basis) or fermionic (anti-symmetric basis) vector oscillators. At critical masses the reduced Weyl zero-form modules become indecomposable. We explicitly project the latter onto the submodules carrying Metsaev's massless representations. The remainder of the reduced system contains a set of Stückelberg fields and dynamical potentials that leads to a smooth flat limit in accordance with the Brink-Metsaev-Vasiliev (BMV) conjecture. In the unitary massless cases in AdS, we identify the Alkalaev-Shaynkman-Vasiliev frame-like potentials and explicitly disentangle their unfolded field equations.

A Basis of Bideterminants for the Coordinate Ring of the Orthogonal Group

We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King–Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isomorphic to the tensor product of orthogonal analogs of left and right Schur modules.

Schur Modules, Weyl Modules, and Capelli Operators

Much work has been done in recent years towards presenting explicitly and with a minimum of machinery the irreducible representations of the general linear group GL(n) over a field K. The representations obtained from the modules named after Schur and Weyl have undergone a tortuous refining process in the hands of notable authors (Akin, Buchsbaum and Weyman [1, 2], Carter and Lusztig [7], Clausen [8, 9], De Concini, Eisenbud and Procesi [10], Green [16], James and Kerber [19], Towber [26, 27]), in the wake of the pioneering work of Schur [23] and Weyl [28]. One objective of such simplifications has been that of obtaining information as to the reducibility of representations when the field K if of positive characteristic, an objective which seems as yet out of reach. The purpose of this work is that of organically presenting the main results on Schur and Weyl modules. Our techniques are at variance with most of those previously used and, we should like to believe, our proofs signify a return to the primordial combinatorial underpinning of the topic. We hope thereby to have attained a maximum of characteristic freeness. Our two warhorses are: (1) the well known straightening technique, first used by Doubilet, Rota and Stein [15], and recently generalized to the supersymmetric case by Grosshans, Rota and Stein [17]. An altogether different approach from which we have also benefitted has been developed by Akin, Buchsbaum and Weyman [1]. Both techniques has been skillfully displayed in the recent work of Kouwenhoven [21]. (2) the use of polarization operators, combined with the striking power of Kostant's Z-form [20], in the guise of De sarme nien, Kung and Rota's Capelli operators [13]. Superalgebraic methods are used in obtaining two distinct standard basis theorems for the case of the exterior letterplace algebra. Such an algebra, with its straightening possibilities, was first considered by Doubilet and Rota [14], and then brought to fruish by Akin, Buchsbaum and Weyman [1]. By these methods we derive what is perhaps the main novelty of the doi:10.1006 aima.1999.1872, available online at http: www.idealibrary.com on

The Space of Triangles, Vanishing Theorems, and Combinatorics

We consider compactifications of (Pn)3\\∪Δij, the space of triples of distinct points in projective space. One such space is a singular variety of configurations of points and lines; another is the smooth compactification of Fulton and MacPherson; and a third is the triangle space of Schubert and Semple.We compute the sections of line bundles on these spaces, and show that they are equal as GL(n) representations to the generalized Schur modules associated to “bad” generalized Young diagrams with three rows (Borel–Weil theorem). On the one hand, this yields Weyl-type character and dimension formulas for the Schur modules; on the other, a combinatorial picture of the space of sections. Cohomology vanishing theorems play a key role in our analysis.

On properties of the Mullineux map with an application to Schur modules

The Mullineux map is an involutory bijection on the set of p-regular partitions of any given integer n, where a partition is called p-regular if no part of it is repeated p or more times. Many combinatorial properties of the Mullineux map make it reasonable to view this map as a p-analogue of the transposition map T on the set of all partitions.Based on the work of Kleshchev [7], it has been shown [2, 5, 14] that the Mullineux map M has the following property.If λ is a p-regular partition of n and Dλ is the p-modular representation of the symmetric group Sn labelled by λ (see [6]) thenformula herewhere sgn is the sign representation of Sn. Thus, from a representation theoretic point of view as well, the Mullineux map is a p-analogue of the transposition map T. The Mullineux map plays a vital role not only in the representation theory of symmetric groups but also in other contexts. The definition of M as given in [9] is quite complicated and to study various questions involving this map it is desirable to have other descriptions. One alternative inductive description of M using the concept of good boxes in a p-regular partition was given by Kleshchev [7] and this was used by Walker to prove a result contained in Theorem 4·5 below; this work was motivated by the investigation of Schur modules. In this paper we study a third description of M based on the operator J on the set of p-regular partitions defined in [13].