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## Modality of representations and geometry of θ-groups

Ch. 16. P. 459-479.

We first establish several general properties of modality of algebraic group

actions. In particular,we introduce the notion of a modality-regular action and prove

that every visible action is modality-regular. Then, using these results, we classify irreducible

linear representations of connected simple algebraic groups of every fixed

modality < 3. Next, exploring a finer geometric structure of linear actions, we generalize

to the case of any cyclically graded semisimple Lie algebra the notion of a

packet (or a Jordan/decomposition class) and establish the properties of packets.

### In book

Kac, V. G., Popov V. Vol. 326. , Copyright Holder Springer Nature Switzerland AG, 2018

Popov V., Algebraic groups whose orbit closures contain only finitely many orbits / Cornell University. Series math "arxiv.org". 2017. No. 1707.06914 [math.AG].

We classify all connected affine algebraic groups G such that there are only finitely many G-orbits in every algebraic G-variety containing a dense open G-orbit. We also prove that G enjoys this property if and only if every irreducible algebraic G-variety X is modality-regular, i.e., the modality of X (in the sense of V. Arnol’d) ...

Added: July 24, 2017

Popov V., , in: Oberwolfach Reports. Vol. 10. Issue 2.: European Mathematical Society Publishing house, 2013.. P. 1053-1055.

The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes. ...

Added: February 26, 2014

Popov V., Number of components of the nullcone / Cornell University. Series math "arxiv.org". 2015. No. 1503.08303.

For every pair (G, V ) where G is a connected simple
linear algebraic group and V is a simple algebraic G-module with
a free algebra of invariants, the number of irreducible components
of the nullcone of unstable vectors in V is found. ...

Added: March 31, 2015

Popov V., Doklady Mathematics 2017 Vol. 96 No. 1 P. 312-314

For connected simple algebraic groups defined over an algebraically closed field of characteristic
zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained. ...

Added: June 30, 2017

Popov V., Around the Abhyankar–-Sathaye conjecture / Cornell University. Series math "arxiv.org". 2014. No. 1409.6330.

A "rational" version of the strengthened form of the Commuting Derivation Conjecture, in which the assumption of commutativity is dropped, is proved. A systematic method of constructing in any dimension greater than 3 the examples answering in the negative a question by M. El Kahoui is developed. ...

Added: September 24, 2014

Воейкова М., Ivanova K., В кн.: Проблемы функциональной грамматики. Предикативные категории в высказывании и целостном тексте.. Т. 6: Предикативные категории в высказывании и целостном тексте.: М.: Издательский дом ЯСК, 2017.. С. 308-331.

The subject of the current chapter is the functions of perfective and imperfective imperatives of give-verbs dajand davaj. It is known that in the Russian language these forms are frequent and possess a wide functional range. Further grammaticalization of these forms shows a divergence, which can not be completely explained by the perfective-imperfective opposition in the imperative ...

Added: October 26, 2018

Feigin E., Makhlin I., Vertices of FFLV polytopes / Cornell University. Series math "arxiv.org". 2016. No. arXiv:1604.08844 .

FFLV polytopes describe monomial bases in irreducible representations of sln. We study various sets of vertices of FFLV polytopes. First, we prove the locality of set of vertices with respect to the type A Dynkin diagram. Second, we describe all the permutation vertices. Third, we describe all the simple vertices and prove that their number ...

Added: May 6, 2016

Braverman A., Rybnikov L. G., Feigin B. L. et al., Communications in Mathematical Physics 2011 Vol. 308 No. 2 P. 457-478

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts ...

Added: May 12, 2012

Popov V., Transformation Groups 2011 Vol. 16 No. 3 P. 827-856

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the ...

Added: March 16, 2013

Avdeev R., Петухов А. В., Математический сборник 2014 Т. 205 № 9 С. 3-48

For every finite-dimensional vector space V and every V-flag variety X we list all connected reductive subgroups in GL(V) acting spherically on X. ...

Added: October 22, 2014

Popov V., On the equations defining affine algebraic groups / Cornell University. Series math "arxiv.org". 2015. No. 1508.02860.

For the coordinate algebras of connected affine algebraic groups, we explore the problem of finding a presentation by generators and relations canonically determined by the group structure. ...

Added: August 13, 2015

Letuchiy A., Oslo studies in language 2012 Vol. 4 No. 1 P. 31-51

In the paper I consider the causative constructions in Russian. I examine the use of tense and aspect in constructions with the verbs zastavit’ / zastavljat’ ‘make’ and pozvolit’ / pozvoljat’ ‘let, allow’. I also include the verb delat’ / sdelat ‘make’ in my analysis, though this verb has special syntactic and semantic characteristics.
The striking ...

Added: October 31, 2012

Lander Y., В кн.: Исследования по теории грамматики 3: Ирреалис и ирреальность. .: М.: Гнозис, 2004.. С. 147-176.

A large number of Pama-Nyungan languages spoken in Australia display the existence of verb forms whose marking involves a dative suffix or a morpheme related to a dative formant. These dative verb forms show an interesting behavior in that they (i) can be used both in independent and in subordinate clauses and (ii) have a ...

Added: May 1, 2014

Arzhantsev I., Celik D., Hausen J., Journal of Algebra 2013 Vol. 387 P. 87-98

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. ...

Added: November 13, 2013

Bershtein M., Gavrylenko P., Marshakov A., Twist-field representations of W-algebras, exact conformal blocks and character identities / . 2017. No. 1705.00957.

We study twist-field representations of the W-algebras and generalize the construction of the corresponding vertex operators to D- and B-series. We demonstrate how the computation of characters of such representations leads to the nontrivial identities involving lattice theta-functions. We propose a construction of their exact conformal blocks, which for D-series express them in terms of ...

Added: May 4, 2017

Popov V., Proceedings of the Steklov Institute of Mathematics, Springer 2019 Vol. 307 P. 193-197

We prove that, for any prime integer $p\geqslant 2$, there exists an algebraic action of the two-dimensional Witt group $W_2(p)$ on an al\-geb\-raic variety $X$ and a point $x\in X$ such that the closure of the $W_2(p)$-orbit of $x$
in $X$ contains infinitely many $W_2(p)$-orbits.\;This is related to the problem of extending from characteristic zero to ...

Added: March 30, 2020

Feigin E., Makedonskyi I., Weyl modules for osp(1,2) and nonsymmetric Macdonald polynomials / Cornell University. Series arXiv "math". 2015. No. 1507.01362.

The main goal of our paper is to establish a connection between the Weyl modules of the current Lie superalgebras (twisted and untwisted) attached to osp(1,2) and the nonsymmetric Macdonald polynomials of types $A_2^2$ and ${A_2}^{2\dagger}$ . We compute the dimensions and construct bases of the Weyl modules. We also derive explicit formulas for the ...

Added: July 8, 2015

Avdeev R., Transformation Groups 2021 Vol. 26 No. 2 P. 403-431

Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...

Added: September 9, 2021

Popov V., Transformation Groups 2021 Vol. 26 No. 2 P. 671-689

We explore connected ane algebraic groups G, which enjoy the following finiteness property (F): for every algebraic action of G, the closure of every G-orbit contains only nitely many G-orbits. We obtain two main results. First, we classify such groups. Namely, we prove that a connected affine algebraic group G enjoys property (F) if and ...

Added: September 21, 2021

Terekhovich V., Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics 2018 Vol. 62 P. 189-201

Despite the importance of the variational principles of physics, there have been relatively few attempts to consider them for a realistic framework. In addition to the old teleological question, this paper continues the recent discussion regarding the modal involvement of the principle of least action and its relations with the Humean view of the laws ...

Added: February 12, 2020

Bezrukavnikov R., Finkelberg M. V., Wreath Macdonald polynomials and categorical McKay correspondence (with Appendices by Ivan Losev, Vadim Vologodsky) / Cornell University. Series math "arxiv.org". 2012. No. 1208.3696.

Mark Haiman has reduced Macdonald positivity conjecture to a statement about geometry of the Hilbert scheme of points on the plane, and formulated a generalization of the conjectures where the symmetric group is replaced by the wreath product $S_n\ltimes (Z/r Z)^n$. He has proven the original conjecture by establishing the geometric statement about the Hilbert ...

Added: February 6, 2013

Smilga I., Construction of Milnorian representations / Cornell University. Series arXiv "math". 2018. No. 1802.07193.

We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let G be a semisimple real Lie group with a representation rho on a finite-dimensional real vector space V, that does not satisfy the criterion from the previous paper. ...

Added: September 26, 2018

Lyashevskaya O., Ovsjannikova M., Szymor N. et al., , in: Quantitative approaches to the Russian language. .: Abingdon: Routledge, 2018.. P. 51-78.

The domain of modality is structurally diverse and may be described in multiple ways (for example, see Perkins, 1983; Wierzbicka, 1987; Hengeveld, 1988/2004; Sweetser, 1990; Bondarko, 1990; Bybee et al., 1994; van der Auwera and Plungian, 1998; Palmer, 2001; Hansen, 2004; Nuyts, 2006; Khrakovsky, 2007). The article reports on the Russian part of a larger survey ...

Added: October 24, 2017

Avdeev R., Cupit-Foutou S., Advances in Mathematics 2018 Vol. 328 P. 1299-1352

Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme M_Γ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an `adjoint torus' T_ad and has a distinguished T_ad-fixed point X_0. ...

Added: February 25, 2018