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Abstract: Abstract Over a field of positive characteristic, a semisimple algebraic group G may have some nonreduced parabolic subgroup P. In this paper, we study the Schubert and Bott–Samelson–Demazure–Hansen (BSDH) varieties of G/P, with P nonreduced, when the base field is perfect. It is shown that, in general, the Schubert and BSDH varieties of such a G/P are not normal, and the projection of the BSDH variety onto the Schubert variety has nonreduced fibers at closed points. When the base field is finite, the generalized convolution morphisms between BSDH varieties (as in [dCHL18]) are also studied. It is shown that the decomposition theorem holds for such morphisms, and the pushforward of intersection complexes by such morphisms are Frobenius semisimple. PubDate: 2022-04-27

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Abstract: Abstract Deligne’s category \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right) \) is a tensor category depending on a parameter t “interpolating” the categories of representations of the symmetric groups Sn. We construct a family of categories Cλ (depending on a vector of variables λ = (λ1, λ2, … , λl), that may be specialised to values in the ground ring) which are module categories over \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right). \) The categories Cλ are defined over any ring and are constructed by interpolating permutation representations. Further, they admit specialisation functors to Sn-mod which are tensor-compatible with the functors \( \underset{\_}{\mathrm{Rep}}\left({S}_t\right)\to {S}_n-\operatorname{mod}. \) We show that Cλ can be presented using the Kostant integral form of Lusztig’s universal enveloping algebra \( \dot{U}\left({\mathfrak{gl}}_{\infty}\right), \) and exhibit a categorification of some stability properties of Kronecker coefficients. PubDate: 2022-04-27

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Abstract: Abstract We compute the group of automorphisms of an arbitrary ind-variety of (possibly isotropic) generalized flags. Such an ind-variety is a homogeneous ind-space for one of the ind-groups \(SL(\infty )\) , \(O(\infty )\) or \(Sp(\infty )\) . We show that the respective automorphism groups are much larger than \(SL(\infty )\) , \(O(\infty )\) or \(Sp(\infty )\) , and present the answer in terms of Mackey groups. The latter are groups of automorphisms of non-degenerate pairings of (in general infinite-dimensional) vector spaces. An explicit matrix form of the automorphism group of an arbitrary ind-variety of generalized flags is also given. The case of the Sato grassmannian is considered in detail, and its automorphism group is the projectivization of the connected component of unity in the group known as Japanese \(GL(\infty )\) . PubDate: 2022-04-25

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Abstract: Abstract We prove that the number of conjugacy classes of a finite group G consisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup of G. This is predicted by the Alperin Weight Conjecture. PubDate: 2022-04-22

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Abstract: Abstract We prove a motivic analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized ag manifold G/B multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure–Lusztig (D–L) operators, identifies an Euler characteristic above to a formula of Brubaker, Bump and Licata for the Iwahori–Whittaker functions of the principal series representation of the p-adic Langlands dual group. As a corollary, we recover the classical Casselman–Shalika formula for the spherical Whittaker function. The proofs are based on localization in equivariant K-theory, and require a geometric interpretation of how the Hecke inverse of a D–L operator acts on the class of a point. We prove that the Hecke inverse operators give Grothendieck–Serre dual classes of the motivic classes, a result which might be of independent interest. In an Appendix jointly authored with Dave Anderson, we show that if the line bundle is trivial, we recover a generalization of a classical formula by Kostant, Macdonald, Shapiro and Steinberg for the Poincaré polynomial of G/B; the generalization we consider is due to Akyıldız and Carrell and replaces G/B by any smooth Schubert variety. PubDate: 2022-04-19

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Abstract: Abstract The twin group TWn on n strands is the group generated by \(t_{1}, \dots , t_{n-1}\) with defining relations \({t_{i}^{2}}=1\) , titj = tjti if i − j > 1. We find a new instance of semisimple Schur–Weyl duality for tensor powers of a natural n-dimensional reflection representation of TWn, depending on a parameter q. At q = 1, the representation coincides with the natural permutation representation of the symmetric group, so the new Schur–Weyl duality may be regarded as a q-analogue of the one motivating the definition of the partition algebra. PubDate: 2022-04-13

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Abstract: Abstract In this article, we shall look into the existence of vertical cylinders contained in a weak del Pezzo fibration as a generalization of the former work of Dubouloz and Kishimoto in which they observed vertical cylinders found in del Pezzo fibrations. With the essence lying in the existence of a cylinder in the generic fiber, we devote ourselves mainly to a geometry of minimal weak del Pezzo surfaces defined over a field of characteristic zero from the point of view of cylinders. As a result, we give the classification of minimal weak del Pezzo surfaces defined over a field of characteristic zero, and we show that weak del Pezzo fibrations containing vertical cylinders are quite restrictive. PubDate: 2022-04-13

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Abstract: Abstract Suppose G is a finite cyclic group and M a closed smooth G-manifold. We will show that there is a nonsingular real algebraic G-variety X that is equivariantly diffeomorphic to M so that all G-vector bundles over X are strongly algebraic. PubDate: 2022-04-11

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Abstract: Abstract Let G be a small finite subgroup of GL(2,ℂ) and let \( \overset{\sim }{\varphi }:{\mathbbm{A}}^2\to {\mathbbm{A}}^2 \) be a G- equivariant étale endomorphism of the affine plane. We show that \( \overset{\sim }{\varphi } \) is an automorphism if the order of G is even. The proof depends on an analysis of a quasi-étale endomorphism φ induced by \( \overset{\sim }{\varphi } \) on the singular quotient surface \( {\mathbbm{A}}^2/G \) whose smooth part X° has the standard \( {\mathbbm{A}}_{\ast}^1 \) -fibration p°: X° → ℙ1. If φ preserves the standard \( {\mathbbm{A}}_{\ast}^1 \) -fibbration p° then both φ and \( \overset{\sim }{\varphi } \) are automorphisms. We look for the condition with which φ preserves the standard \( {\mathbbm{A}}_{\ast}^1 \) -fibration and prove as a consequence that \( \overset{\sim }{\varphi } \) is an automorphism if G has even order. PubDate: 2022-04-08

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Abstract: Abstract Let k be an algebraically closed field of characteristic p > 2. We compute the Weyl filtration multiplicities in indecomposable tilting modules and the decomposition numbers for the symplectic group over k in terms of cap-curl diagrams under the assumption that p is bigger than the greatest hook length in the largest partition involved. As a corollary we obtain the decomposition numbers for the Brauer algebra under the same assumptions. Our work combines ideas from work of Cox and De Visscher and work of Shalile with techniques from the representation theory of reductive groups. PubDate: 2022-04-08

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Abstract: Abstract The paper contains results that characterize the Donkin–Koppinen filtration of the coordinate superalgebra K[G] of the general linear supergroup G = GL(m n) by its subsupermodules CΓ = OΓ(K[G]). Here, the supermodule CΓ is the largest subsupermodule of K[G] whose composition factors are irreducible supermodules of highest weight ⋋, where ⋋ belongs to a finitely-generated ideal Γ of the poset X(T)+ of dominant weights of G. A decomposition of G as a product of subsuperschemes U–×Gev×U+ induces a superalgebra isomorphism ϕ* K[U–]⊗K[Gev]⊗K[U+]≃K[G]. We show that CΓ=ϕ*(K[U–]⊗MΓK[U+]), where MΓ=OΓ(K[Gev]). Using the basis of the module MΓ, given by generalized bideterminants, we describe a basis of CΓ. Since each CΓ is a subsupercoalgebra of K[G], its dual \( {C}_{\Gamma}^{\ast }={S}_{\Gamma} \) is a (pseudocompact) superalgebra called the generalized Schur superalgebra. There is a natural superalgebra morphism πΓ : Dist(G) → SΓ such that the image of the distribution algebra Dist(G) is dense in SΓ. For the ideal \( X{(T)}_l^{+}, \) of all weights of fixed length l, the generators of the kernel of \( {\uppi}_{X{(T)}_l^{+}} \) are described. PubDate: 2022-03-31

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Abstract: Abstract We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, because of the Iwasawa decomposition G = KAU+, the embedding K ,↪ G is a weak homotopy equivalence, in particular π1(G) = π1(K). It thus suffices to determine π1(K), which we achieve by investigating the fundamental groups of generalized ag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized ag variety is a CW decomposition- in particular, we cover the complete symmetrizable situation; furthermore, the results concerning only the structure of π1(K) actually also hold in the nonsymmetrizable two-spherical case. PubDate: 2022-03-31

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Abstract: Abstract We find a necessary condition for the existence of an action of a Lie group G by quaternionic automorphisms on an integrable quaternionic manifold in terms of representations of ð”¤. We check this condition and prove that a Riemannian symmetric space of dimension 4n for n ≥ 2 has an invariant integrable almost quaternionic structure if and only if it is quaternionic vector space, quaternionic hyperbolic space, or quaternionic projective space. PubDate: 2022-03-31

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Abstract: Abstract We show that not every Salem number appears as the growth rate of a cocompact hyperbolic Coxeter group. We also give a new proof of the fact that the growth rates of planar hyperbolic Coxeter groups are spectral radii of Coxeter transformations, and show that this need not be the case for growth rates of hyperbolic tetrahedral Coxeter groups. PubDate: 2022-03-31

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Abstract: Abstract We define the equivariant Cox ring of a normal variety with algebraic group action. We study algebraic and geometric aspects of this object and show how it is related to the ordinary Cox ring. Then we specialize to the case of normal rational varieties of complexity one under the action of a connected reductive group G. We show that the G-equivariant Cox ring is then a finitely generated integral normal G-algebra. Under a mild additional condition, we give a presentation of its subalgebra of U-invariants, where U is the unipotent part of a Borel subgroup of G. The ordinary Cox ring is also finitely generated and inherits a canonical structure of U-algebra. Using a work of Hausen and Herppich, we prove that the subalgebra of U-invariants of the Cox ring is a finitely generated Cox ring of a variety of complexity one under the action of a torus. As an application, we provide a criterion of combinatorial nature for the Cox ring of an almost homogeneous G-variety of complexity one to have log terminal singularities. Finally, we prove that for a normal rational G-variety of complexity one satisfying a mild additional condition (e.g., complete or almost homogeneous), the iteration of Cox rings is finite. PubDate: 2022-03-31

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Abstract: Abstract To each complex semisimple Lie algebra \( \mathfrak{g} \) and regular element a ∈ \( \mathfrak{g} \) reg, one associates a Mishchenko–Fomenko subalgebra \( \mathcal{F} \) a ⊆ ℂ[ \( \mathfrak{g} \) ]. This subalgebra amounts to a completely integrable system on the Poisson variety \( \mathfrak{g} \) , and as such has a bifurcation diagram Σa ⊆ Spec( \( \mathcal{F} \) a). We prove that Σa has codimension one in Spec( \( \mathcal{F} \) a) if a ∈ \( \mathfrak{g} \) reg is not nilpotent, and that it has codimension one or two if a ∈ \( \mathfrak{g} \) reg is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of Σa. PubDate: 2022-03-31

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Abstract: Abstract We define a set of PBW-semistandard tableaux that is in a weight-preserving bijection with the set of monomials corresponding to integral points in the Feigin–Fourier–Littelmann–Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the multihomogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety with respect to the Plücker embedding. These relations consist of type Α degenerate Plücker relations and a set of degenerate linear relations that we obtain from De Concini’s linear relations. PubDate: 2022-03-31

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Abstract: Abstract We develop a theory of two-parameter quantum polynomial functors. Similar to how (strict) polynomial functors give a new interpretation of polynomial representations of the general linear groups GLn, the two-parameter polynomial functors give a new interpretation of (polynomial) representations of the quantum symmetric pair ( \( {U}_{Q,q}^B \) ( \( \mathfrak{gl} \) n), Uq( \( \mathfrak{gl} \) n)) which specializes to type AIII/AIV quantum symmetric pairs. The coideal subalgebra \( {U}_{Q,q}^B \) ( \( \mathfrak{gl} \) n) appears in a Schur–Weyl duality with the type B Hecke algebra \( {\mathcal{H}}_{Q,q}^B \) (d). We endow two-parameter polynomial functors with a cylinder braided structure which we use to construct the two-parameter Schur functors. Our polynomial functors can be precomposed with the quantum polynomial functors of type A producing new examples of action pairs. PubDate: 2022-03-31

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Abstract: Abstract We study the simple Bershadsky–Polyakov algebra ð’²k = ð’²k(sl3, fθ) at positive integer levels and classify their irreducible modules. In this way, we confirm the conjecture from [9]. Next, we study the case k = 1. We discover that this vertex algebra has a Kazama–Suzuki-type dual isomorphic to the simple affine vertex superalgebra Lk′ (osp(1 2)) for k′ = –5=4. Using the free-field realization of Lk′ (osp(1 2)) from [3], we get a free-field realization of ð’²k and their highest weight modules. In a sequel, we plan to study fusion rules for ð’²k. PubDate: 2022-03-29

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Abstract: Abstract We consider the PBW basis of the quantum toroidal algebra of \({\mathfrak {g}\mathfrak {l}}_{n}\) , which was developed in Neguț (Adv. Math. 372, 2020), and prove commutation relations between its generators akin to the ones studied in Burban and Schiffmann (Duke Math. J. 161(7):1171–1231, 2012) for n = 1. This gives rise to a new presentation of the quantum toroidal algebra of type A. PubDate: 2022-03-25