We develop a theory of parabolic induction and restriction functors relating modules over Coulomb branch algebras, in the sense of Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof’s induction and restriction functors for Cherednik algebras, but their definition uses different tools.

After this general definition, we focus on quiver gauge theories attached to a quiver $\Gamma $. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak {g}_{\Gamma }$ on category $ \mathcal {O}$ for these Coulomb branch algebras. When $ \Gamma $ is of Dynkin type, the Coulomb branch algebras are truncated shifted Yangians and quantize generalized affine Grassmannian slices. Thus, we regard our action as a categorification of the geometric Satake correspondence.

To establish this categorical action, we define a new class of ‘flavoured’ KLRW algebras, which are similar to the diagrammatic algebras originally constructed by the second author for the purpose of tensor product categorification. We prove an equivalence between the category of Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a flavoured KLRW algebra. This equivalence relates the categorical action by induction and restriction functors to the usual categorical action on modules over a KLRW algebra.

Let $M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ denote the matrix multiplication tensor (and write $M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ ), and let $\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ denote the determinant polynomial considered as a tensor. For a tensor T, let $\underline {\mathbf {R}}(T)$ denote its border rank. We (i) give the first hand-checkable algebraic proof that $\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ , (ii) prove $\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ and $\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was $M_{\langle 2\rangle }$ , (iii) prove $\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , (iv) prove $\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , improving the previous lower bound of $12$ , (v) prove $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ for all $\mathbf {n}\geq 25$ , where previously only $\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ was known, as well as lower bounds for $4\leq \mathbf {n}\leq 25$ , and (vi) prove $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ for all $\mathbf {n} \ge 18$ , where previously only $\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.

The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensor T and an integer r, in a finite number of steps, either outputs that there is no border rank r decomposition for T or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable when T has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory.

An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group ${\mathbb {K}^{*}}$ commuting with the G-action. We show that X is determined by the ${\mathbb {K}^{*}}$-variety $X^U$ of fixed points under a maximal unipotent subgroup $U \subset G$. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient $X /\!\!/ G$.

If G is of type ${\mathsf {A}_n}$ ($n\geq 2$), ${\mathsf {C}_{n}}$, ${\mathsf {E}_{6}}$, ${\mathsf {E}_{7}}$, or ${\mathsf {E}_{8}}$, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If $n \geq 5$, every smooth affine $\operatorname {\mathrm {SL}}_n$-variety of dimension $< 2n-2$ is an $\operatorname {\mathrm {SL}}_n$-vector bundle over the smooth quotient $X /\!\!/ \operatorname {\mathrm {SL}}_n$, with fiber isomorphic to the natural representation or its dual.

L'analyse de la démonstration par contradiction de Frécon 2018 qui est faite dans Poizat 2018 met en évidence la structure symétrique des groupes de rang de Morley fini sans involutions; en effet, cette démonstration consiste en la construction d'un espace symétrique de dimension deux (“un plan”), puis à montrer que ce plan ne peut exister.

Aux sous-espaces symétriques définissables de ces groupes sont associées des symétries et des transvexions, qu'on entreprend d'étudier ici dans l'abstrait, sans référence à un groupe qui les enveloppe; cela nous mène à considérer des structures introduites axiomatiquement que nous appelons symétrons (plutôt qu'ensembles symétriques diadiques, comme les ont nommées Lawson & Lim 2004).

Le $Z^*$ -Theorem de Glauberman permet d'élucider complètement la structure des symétrons finis: chacun est isomorphe à l'ensemble des symétries associées à un sous-espace symétrique d'un groupe fini sans involutions, qui est loin d'être uniquement déterminé: de fait, il existe des groupes finis non isomorphes qui ont les mêmes symétries, et aussi des symétrons finis qui ne sont pas isomorphes aux symétries d'un groupe,

La situation est plus incertaine dans le cas des symétrons de rang de Morley fini, ou même algébriques, qui sont l'objet d'étude principal de cet article. Mais bien qu'un symétron soit une structure nettement plus faible qu'un groupe, nous pouvons étendre aux symétrons des résultats bien connus à propos des groupes de rang de Morley fini: condition de chaîne, décomposition en composantes connexes, caractérisation des parties définissables génériques, génération elliptique, etc. Ces propriétés sont nouvelles même dans le cas des sous-espaces symétriques d’un groupe, et permettent de court-circuiter les calculs de Frécon dans la construction de son plan paradoxal.

En outre, sous l'hypothèse de la Conjecture d'Algébricité, nous généralisons le Théorème de Glauberman au contexte de rang de Morley fini.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$ , where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

This paper is dedicated to a problem raised by Jacquet Tits in 1956: the Weyl group of a Chevalley group should find an interpretation as a group over what is nowadays called $\mathbb{F}_{1}$, the field with one element. Based on Part I of The geometry of blueprints, we introduce the class of Tits morphisms between blue schemes. The resulting Tits category$\text{Sch}_{{\mathcal{T}}}$ comes together with a base extension to (semiring) schemes and the so-called Weyl extension to sets. We prove for ${\mathcal{G}}$ in a wide class of Chevalley groups—which includes the special and general linear groups, symplectic and special orthogonal groups, and all types of adjoint groups—that a linear representation of ${\mathcal{G}}$ defines a model $G$ in $\text{Sch}_{{\mathcal{T}}}$ whose Weyl extension is the Weyl group $W$ of ${\mathcal{G}}$. We call such models Tits–Weyl models. The potential of Tits–Weyl models lies in (a) their intrinsic definition that is given by a linear representation; (b) the (yet to be formulated) unified approach towards thick and thin geometries; and (c) the extension of a Chevalley group to a functor on blueprints, which makes it, in particular, possible to consider Chevalley groups over semirings. This opens applications to idempotent analysis and tropical geometry.

Let G be a reductive algebraic group over a field k, such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains (Gm)2. Let K1G be the non-stable K1-functor associated to G, also called the Whitehead group of G. We show that K1G(k) = K1G (k[X1 ,…, Xn]) for any n ≥ 1. If k is perfect, this implies that K1G (R) = K1G (R[X]) for any regular k-algebra R. If k is infinite perfect, one also deduces that K1G (R) → K1G (K) is injective for any local regular k-algebra R with the fraction field K.

We compute the equation of the 7-secant variety to the Veronese variety (P4,O(3)), its degree is 15. This is the last missing invariant in the Alexander-Hirschowitz classification. It gives the condition to express a homogeneous cubic polynomial in 5 variables as the sum of 7 cubes (Waring problem). The interesting side in the construction is that it comes from the determinant of a matrix of order 45 with linear entries, which is a cube. The same technique allows to express the classical Aronhold invariant of plane cubics as a pfaffian.

We derive the spectrum of the Laplace–Beltrami operator on the quotient orbifold of the nonhyperbolic triangle groups.

Let $G$ be a classical algebraic group, $X$ a maximal rank reductive subgroup and $P$ a parabolic subgroup. This paper classifies when $X\backslash G/P$ is finite. Finiteness is proven using geometric arguments about the action of $X$ on subspaces of the natural module for $G$. Infiniteness is proven using a dimension criterion that involves root systems.

We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.

This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G\left( E \right)$, with periods by a subgroup $G\left( F \right)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E=F\oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy.

The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms—with non vanishing periods—and the resulting bi-period distributions associated to “periodic” automorphic forms, are related to analogous bi-period distributions associated to “periodic” automorphic forms on the endoscopic symmetric spaces $H\left( E \right)/H\left( F \right)$. This offers a sharpening of the theory of liftings, where periods play a key role.

The stabilization depends on the “fundamental lemma”, which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a “transfer factor”. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case.

Finally, the fundamental lemma is verified for $\text{SL}\left( 2 \right)$.

The u-invariant u(K) of a field K is the smallest number such that every k-dimensional regular quadratic form over K is universal. Let O(V,f) be the orthogonal group of a finite-dimensional regular metric vector space over a field K of characteristic distinct from 2. Let π ∊ 0(V), B(π) := V(π - 1), dim[B(π) ∩ kernel(π - 1)] > u(K). Given λ1,..., λm ∊ K* where m := dim B(π) - u(K) + 1. Then π = σ1······σk where k := dim B(π) and σi is a symmetry with negative space Kai and f(ai, ai) = λi for i ∊ { 1 , . . . , m}. We prove similar theorems also for symplectic groups where transvections are taken as generators.