## Kirillov, Anatol

Ì¾Á° Kirillov, Anatoli
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E-Mail kirillov ¡Êemail¥¢¥É¥ì¥¹¤Ë¤Ï¡÷kurims.kyoto-u.ac.jp ¤ò¤Ä¤±¤Æ¤¯¤À¤µ¤¤¡Ë
¸¦µæÆâÍÆ Algebraic Analysis and Algebraic Combinatorics
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¡¡In recent years (2007-2012) 10 my papers (according to Mat.Sci.Net) have been published in different reviewed Mathematical journals, and 5 my papers are still in press. In these papers different aspects of Classical and Quantum Schubert Calculi, Algebraic Combinatorics, Discrete Integrable Systems and Representation Theory have been investigated.
¡¡In papers [3,4,5,6,11] jointly with T.Maeno, we continue the study of various interesting connections between certain noncommutative braided Hopf algebras and Classical and Quantum Schubert and Grothendieck Calculi, as well as noncommutative differential Geometry on Coxeter groups in the sense of S.Majid. In particular,
¡¡we described relations between flat connections on Weyl groups of classical type;
¡¡we find the so-called Nichols--Woronowicz algebra model for the small quantum cohomology ring and the Grothendieck ring of flag varieties of classical type.
¡¡As an application of our approach, we proved positivity of certain $B_n$ type Littlewood--Richardson numbers.
¡¡We extend part of our results to the case of the complex reflection groups, including construction of Dunkl elements and Nichols --Woronowicz type model for the coivariant algebra of the corresponding complex reflection group.
¡¡We have describe relations between elliptic Dunkl elements, which can be treated as a description of the elliptic cohomology of the type A flag varieties.
¡¡In paper [2] I continue the study of certain quadratic algebras related with Schubert Calculus, hyperplane arrangements, double coinvariants and so on. In particular, I computed the Hilbert series of the classical Yang-Baxter algebras, described commutative subalgebras generated by Dunkl's elements, introduced and investigated double Orlik-Solomon algebras and so on. I also constructed the hyper-elliptic representation of the algebra I'm interested in, and relate the former with the theory of elliptic
¡¡hypergeometric functions.
¡¡In paper [1] jointly with B.Feigin and S.Loktev, we find interesting connections between Higher Level Weyl Modules and the theory of plane partitions. In particular, we proved that a certain level k 2D Weyl module has dimension at least equals (conjecturally equals) to the number of plane partitions of generalized staircase shape.
¡¡In papers [7], [8], jointly with R. Sakamoto we began to study different aspects of relationships between the rigged configurations bijection, invented by the author in 1985, and combinatorics of Ball-Box systems, Macdonald polynomials, energy functions and melting crystal models.
¡¡In papers [9], [10], jointly with H.Katsura, N.Kawashima, V.Korepin and S.Tanaka we began to study some statistical models on graphs related with certain problems in condensed matter.
¡¡In papers [12], [13] I have discovered some interesting and new connections between $\beta$-Grothendieck polynomials, $k$-dissections of a convex $(n+k+1)$-gon, Chan-Robbins polytope and some symmetry classes of plane partitions.
1. Combinatorics and Geometry of Higher Level Weyl Modules (with B.Feigin and S.Loktev), Advances in the Matematical Sciences, Ser.2, 221 (2007), 33-48.
2. On some quadratic algebras,II, Preprint, 150p., submitted.
3. Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties (with T. Maeno), Algebra and Analysis, 21 (2010) no. 4.
4. Nichols--Woronowicz model of coinvariant algebra of complex reflection groups (with T.Maeno), J. Pure Appl. Algebra, 214 (2010), no. 4, 402-409.
5. Braided differential structure on Weyl groups, quadratic algebras, and elliptic functions (with T. Maeno), Int. Math. Res. Not. IMRN 2008, no. 14.
6. Skew divided difference operators and Schubert polynomials. SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), Paper 072, 14 pp.
7. Paths and Kostka--Macdonald polynomials (with R. Sakamoto), Moscow Mathematical Journal, 9 (2009) no.4, 823-854.
8. Relationships between two approaches: rigged configurations and 10-eliminations, Lett. Math. Phys., 89 (2009), no. 1, 51--65. 37 (with R. Sakamoto).
9. The valence bond solid in quasicrystals (with V. Korepin), 2009, submitted.
10. Entanglement in Valence-Bond-Solid states on symmetric graphs, with H.Katsura, N.Kawashima, V.Korepin and S.Tanaka, Journal Phys. A, 43 (2010), no.255303 28p.
11. Affine nil-Hecke algebras and braided differential structure on affine Weyl groups (with T. Maeno), submitted. Preprint arXiv:1008.3593.
12. Algebraic and combinatorial properties of Dunkl elements, Quantum Integrable Systems, Singapore-2011, World Scientific, 2012.
13. Combinatorial and algebraic properties of Dunkl elements, preprint, 44 pp., submitted.