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Regular version of the site
Of all publications in the section: 40
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Article
Polishchuk A. Selecta Mathematica, New Series. 2018. Vol. 24. No. 1. P. 563-589.
Added: May 2, 2018
Article
Feigin B. L., Makhlin I. Selecta Mathematica, New Series. 2016. Vol. 22. No. 3. P. 1703-1747.

We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type (Formula presented.), i.e., corresponding to the affine Lie algebra (Formula presented.). Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula. © 2016 Springer International Publishing

Added: May 4, 2016
Article
Verbitsky M. Selecta Mathematica, New Series. 2012. Vol. 18. No. 3. P. 539-555.

A knot space in a manifold M is a space of oriented immersions S1M up to Diff(S 1). J.-L. Brylinski has shown that a knot space of a Riemannian threefold is formally Kähler. We prove that a space of knots in a holonomy G 2 manifold is formally Kähler.

Added: Oct 25, 2012
Article
Gorsky Evgeny. Selecta Mathematica, New Series. 2013. Vol. 19. No. 1. P. 125-140.

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of the symmetric group Sn . We compare certain multiplicity spaces in its decomposition into irreducible representations of Sn with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity x^my^n .

Added: Dec 9, 2014
Article
Positselski L. Selecta Mathematica, New Series. 2016.

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Št’ovíček with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notion of a relative dualizing complex of bimodules for a pair of noncommutative ring homomorphisms (Formula presented.) and (Formula presented.), and obtain an equivalence between the R / A-semicoderived category of R-modules and the S / B-semicontraderived category of S-modules. For a homomorphism of commutative rings (Formula presented.), we also construct a tensor structure on the R / A-semicoderived category of R-modules. A vision of semi-infinite algebraic geometry is discussed in the introduction.

Added: Dec 3, 2016
Article
Feigin B. L., Kedem R., Loktev S. et al. Selecta Mathematica, New Series. 2002. Vol. 8. No. 3. P. 419-474.
Added: May 31, 2010
Article
Danilov V., Karzanov A., G.A.Koshevoy. Selecta Mathematica, New Series. 2017. Vol. 23. No. 2. P. 1175-1203.

In 1998, Leclerc and Zelevinsky introduced the notion of weakly separated collections of subsets of the ordered n-element set [n] (using this notion to give a combinatorial characterization for quasi-commuting minors of a quantum matrix). They conjectured the purity of certain natural domains D⊆2[n]D⊆2[n] (in particular, of the hypercube 2[n]2[n] itself, and the hyper-simplex {X⊆[n]:|X|=m}{X⊆[n]:|X|=m} for m fixed), where DD is called pure if all maximal weakly separated collections in DD have the same cardinality. These conjectures have been answered affirmatively. In this paper, generalizing those earlier results, we reveal wider classes of pure domains in 2[n]2[n]. This is obtained as a consequence of our study of a novel geometric–combinatorial model for weakly separated set-systems, so-called combined (polygonaltilingson a zonogon, which yields a new insight in the area.

Added: Oct 5, 2018
Article
Alexander Kuznetsov, Perry A. Selecta Mathematica, New Series. 2017. Vol. 23. No. 1. P. 389-423.

Given a variety Y with a rectangular Lefschetz decomposition of its derived category, we consider a degree n cyclic cover X→Y ramified over a divisor Z⊂Y. We construct semiorthogonal decompositions of Db(X) and Db(Z) with distinguished components AX and AZ and prove the equivariant category of AX (with respect to an action of the nth roots of unity) admits a semiorthogonal decomposition into n−1n−1 copies of AZ. As examples, we consider quartic double solids, Gushel–Mukai varieties, and cyclic cubic hypersurfaces.

Added: Jun 14, 2016
Article
Khoroshkin S. M., Nazarov M. Selecta Mathematica, New Series. 2009. Vol. 14. No. 2. P. 321.
Added: Oct 15, 2012
Article
Feigin B. L., Finkelberg M. V., Rybnikov L. G. et al. Selecta Mathematica, New Series. 2011. Vol. 17. No. 2. P. 337-361.

Laumon moduli spaces are certain smooth closures of the moduli spaces of maps from the projective line to the flag variety of GLn. We calculate the equivariant cohomology rings of the Laumon moduli spaces in terms of Gelfand-Tsetlin subalgebra of U(gln), and formulate a conjectural answer for the small quantum cohomology rings in terms of certain commutative shift of argument subalgebras of U(gln).

Added: Oct 9, 2012
Article
Feigin E., Makedonskyi I. Selecta Mathematica, New Series. 2017. Vol. 23. No. 4. P. 2863-2897.

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr–Shimozono formula in order to prove that the $t=\infty$ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.

 

Added: Oct 10, 2017
Article
Kuznetsov A. G., Shinder E. Selecta Mathematica, New Series. 2018. Vol. 24. No. 4. P. 3475-3500.

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection X of three quadrics in P5 and the corresponding double cover Y→P2 branched over a sextic curve. We show that as soon as the natural Brauer class on Y vanishes, so that X and Y are derived equivalent, the difference [X]−[Y] is annihilated by the affine line class.

Added: Jun 14, 2017
Article
Feigin E. Selecta Mathematica, New Series. 2012. Vol. 18. No. 3. P. 513-537.

Let Fλ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module Vλ. We define a flat degeneration Fλa, which is a GaM variety. Moreover, there exists a larger group Ga acting on Fλa, which is a degeneration of the group G. The group Ga contains GaM as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedded into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of Fλa is generated by the set  of degenerate Plüker relations. We prove that the coordinate ring of Fλa is isomorphic to a direct sum of dual PBW-graded g-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableaux.

Added: Aug 31, 2012
Article
Neguț A., Gorsky E. Selecta Mathematica, New Series. 2017. Vol. 23. No. 3. P. 1909-1930.

We study the Maulik-Okounkov K-theoretic stable basis for the Hilbert scheme of points on the plane, and its dependence of the slope parameter.

Added: Jun 6, 2017
Article
Mirkovic I., Finkelberg M. V., Braverman A. et al. Selecta Mathematica, New Series. 2002. Vol. 8. No. 3. P. 381-418.
Added: Jun 11, 2010
Article
Finkelberg M. V., Ginzburg V., Ionov A. et al. Selecta Mathematica, New Series. 2016. Vol. 22. No. 4. P. 2491-2534.

We study the natural Gieseker and Uhlenbeck compactifications of the rational Calogero–Moser phase space. The Gieseker compactification is smooth and provides a small resolution of the Uhlenbeck compactification. We use the resolution to compute the stalks of the IC-sheaf of the Uhlenbeck compactification.

Added: Sep 4, 2016
Article
Bufetov A., Петров Л. Selecta Mathematica, New Series. 2015. Vol. 21. No. 4. P. 1271-1338.
Asymptotic representation theory of general linear groups GL(n,q) over a finite field leads to studying probability measures \rho on the group U of all infinite uni-uppertriangular matrices over F_q, with the condition that \rho is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points was conjectured by Kerov in connection with nonnegative specializations of Hall-Littlewood symmetric functions. Vershik and Kerov also conjectured the Law of Large Numbers for random Young diagrams distributed according to these measures. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson-Schensted-Knuth (RSK) insertion algorithm which samples random Young diagrams \lambda(n) coming from ergodic measures.
Added: Nov 29, 2015
Article
Benett M., Berenstein A., Chari V. et al. Selecta Mathematica, New Series. 2014. No. 2. P. 585-607.

We study the category of graded representations with finite--dimensional graded pieces for the current algebra associated to a simple Lie algebra. This category has many similarities with the category O of modules for g and in this paper, we use the combinatorics of Macdonald polynomials to prove an analogue of the famous BGG duality in the case of sl(n+1)

Added: Sep 29, 2012
Article
Ginzburg V., Travkin R., Finkelberg M. V. Selecta Mathematica, New Series. 2009. Vol. 14. No. 3-4. P. 607-628.

We compute the Frobenius trace functions of mirabolic character sheaves defined over a finite field. The answer is given in terms of the character values of general linear groups over the finite field, and the structure constants of multiplication in the mirabolic Hall–Littlewood basis of symmetric functions, introduced by Shoji.

Added: Jan 25, 2013
Article
Rovinsky M. Selecta Mathematica, New Series. 2009. No. 15. P. 343-376.
Added: Oct 11, 2011
Article
Losev Ivan, Shelley-Abrahamson S. Selecta Mathematica, New Series. 2018. Vol. 24. No. 2. P. 1729-1804.

For a complex reflection group W with reflection representation hh, we define and study a natural filtration by Serre subcategories of the category O_c(W,h) of representations of the rational Cherednik algebra H_c(W,h). This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov–Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori–Hecke algebras. This yields a method for counting the number of irreducible objects in O_c(W,h) of given support. We apply these techniques to count the number of irreducible representations in O_c(W,h) of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of O_c(W,h) for exceptional Coxeter groups W in many new cases.

Added: Aug 27, 2018
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