Kronecker positivity and 2-modular representation theory

This paper consists of two prongs. Firstly, we prove that any Specht module labelled by a 2-separated partition is semisimple and we completely determine its decomposition as a direct sum of graded simple modules. Secondly, we apply these results and other modular representation theoretic techniques on the study of Kronecker coefficients and hence verify Saxl’s conjecture for several large new families of partitions. In particular, we verify Saxl’s conjecture for all irreducible characters of S n \mathfrak {S}_n which are of 2-height zero.

Rank varieties and 𝜋-points for elementary supergroup schemes

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ⩾ 3 p\geqslant 3 , starting with a definition of a π \pi -point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π \pi -points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k [ t , τ ] / ( t p − τ 2 ) k[t,\tau ]/(t^p-\tau ^2) , where t t has even degree and τ \tau has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme.

The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems

This article deals with the Lipschitz regularity of the “approximate” minimizers for the Bolza type control functional of the form \[ J t ( y , u ) ≔ ∫ t T Λ ( s , y ( s ) , u ( s ) ) d s + g ( y ( T ) ) J_t(y,u)≔\int _t^T\Lambda (s,y(s), u(s))\,ds+g(y(T)) \] among the pairs ( y , u ) (y,u) satisfying a prescribed initial condition y ( t ) = x y(t)=x , where the state y y is absolutely continuous, the control u u is summable and the dynamic is controlled-linear of the form y ′ = b ( y ) u y’=b(y)u . For b ≡ 1 b\equiv 1 the above becomes a problem of the calculus of variations. The Lagrangian Λ ( s , y , u ) \Lambda (s,y,u) is assumed to be either convex in the variable u u on every half-line from the origin (radial convexity in u u ), or partial differentiable in the control variable and satisfies a local Lipschitz regularity on the time variable, named Condition (S). It is allowed to be extended valued, discontinuous in y y or in u u , and non convex in u u . We assume a very mild growth condition, actually a violation of the Du Bois-Reymond–Erdmann equation for high values of the control, that is fulfilled if the Lagrangian is coercive as well as in some almost linear cases. The main result states that, given any admissible pair ( y , u ) (y,u) , there exists a more convenient admissible pair ( y ¯ , u ¯ ) (\overline y, \overline u) for J t J_t where u ¯ \overline u is bounded, y ¯ \overline y is Lipschitz, with bounds and ranks that are uniform with respect to t , x t,x in the compact subsets of [ 0 , T [ × R n [0,T[\times \mathbb {R}^n . The result is new even in the superlinear case. As a consequence, there are minimizing sequences that are formed by pairs of equi-Lipschitz trajectories and equi-bounded controls. A new existence and regularity result follows without assuming any kind of Lipschitzianity in the state variable. We deduce, without any need of growth conditions, the nonoccurrence of the Lavrentiev phenomenon for a wide class of Lagrangians containing those that satisfy Condition (S), are bounded on bounded sets “well” inside the effective domain and are radially convex in the control variable. The methods are based on a reparametrization technique and do not involve the Maximum Principle.

Characterizations of monadic NIP

We give several characterizations of when a complete first-order theory T T is monadically NIP, i.e. when expansions of T T by arbitrary unary predicates do not have the independence property. The central characterization is a condition on finite satisfiability of types. Other characterizations include decompositions of models, the behavior of indiscernibles, and a forbidden configuration. As an application, we prove non-structure results for hereditary classes of finite substructures of non-monadically NIP models that eliminate quantifiers.

What makes a complex a virtual resolution?

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’s vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use L 2 L^{2} -harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

Overgroups of regular unipotent elements in simple algebraic groups

We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.

Tilting modules, dominant dimensions and Brauer-Schur-Weyl duality

In this paper we use the dominant dimension with respect to a tilting module to study the double centraliser property. We prove that if A A is a quasi-hereditary algebra with a simple preserving duality and T T is a faithful tilting A A -module, then A A has the double centralizer property with respect to T T . This provides a simple and useful criterion which can be applied in many situations in algebraic Lie theory. We affirmatively answer a question of Mazorchuk and Stroppel by proving the existence of a unique minimal basic tilting module T T over A A for which A = E n d E n d A ( T ) ( T ) A=End_{End_A(T)}(T) . As an application, we establish a Schur-Weyl duality between the symplectic Schur algebra S K s y ( m , n ) S_K^{sy}(m,n) and the Brauer algebra B n ( − 2 m ) \mathfrak {B}_n(-2m) on the space of dual partially harmonic tensors under certain condition.

Characteristic-free test ideals

Tight closure test ideals have been central to the classification of singularities in rings of characteristic p > 0 p>0 , and via reduction to characteristic p > 0 p>0 , in equal characteristic 0 as well. Their properties and applications have been described by Schwede and Tucker [Progress in commutative algebra 2, Walter de Gruyter, Berlin, 2012]. In this paper, we extend the notion of a test ideal to arbitrary closure operations, particularly those coming from big Cohen-Macaulay modules and algebras, and prove that it shares key properties of tight closure test ideals. Our main results show how these test ideals can be used to give a characteristic-free classification of singularities, including a few specific results on the mixed characteristic case. We also compute examples of these test ideals.