BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anton Nazarov (Saint Petersburg State University) DTSTART:20230113T090000Z DTEND:20230113T103000Z DTSTAMP:20260423T024758Z UID:BIMSA-ISS/1 DESCRIPTION:Title: Skew Howe duality\, limit shapes of Young diagrams and universal fluctua tions\nby Anton Nazarov (Saint Petersburg State University) as part of BIMSA Integrable Systems Seminar\n\n\nAbstract\nSchur-Weyl\, Howe and ske w Howe dualities in representation theory of groups lead to multiplicity-f ree decompositions of certain spaces into irreducible representations and can be used to introduce probability measures on Young diagrams that param eterize irreducible representations. It is interesting to study the behavi or of such measures in the limit\, when groups become infinite or infinite -dimensional. Schur-Weyl duality and GL(n)-GL(k) Howe duality are related to classical works of Anatoly Vershik and Sergey Kerov\, as well as Logand -Schepp\, Cohn-Larsen-Propp and Baik-Deift-Johannson. Skew GL(n)-GL(k) How e duality was considered by Gravner\, Tracy and Widom\, who were intereste d in the local fluctuations of the diagrams\, the limit shapes were studie d Sniady and Panova. They demonstrated that results by Romik and Pittel on limit shapes of rectangular Young tableaux are applicable in this case.\n We consider skew Howe dualities for the actions of classical Lie group pai rs: GL(n)-GL(k)\, Sp(2n)-Sp(2k)\, SO(2n)-O(2k) on the exterior algebras. W e describe explicitly the limit shapes for probability measures defined by the ratios of dimensions and demonstrate that they are essentially the sa me for all classical Lie groups. Using orthogonal polynomials we prove cen tral limit theorem for global fluctuations around these limit shapes. Usin g free-fermionic representation we study local fluctuations for more gener al measures given by ratios of representation characters for skew GL(n)-GL (k) Howe duality. These fluctuations are described by Tracy-Widom distribu tion in the generic case and in the corner by a certain discrete distribut ion\, first obtained in papers by Gravner\, Tracy and Widom. Study of loca l fluctuations for other classical series remains an open problem\, but we present numerical evidence that these distributions are universal.\n\nBas ed on joint works with Dan Betea\, Pavel Nikitin\, Olga Postnova\,\nDaniil Sarafannikov and Travis Scrimshaw. See arXiv:2010.16383\,\n2111.12426\, 2 208.10331\, 2211.13728.\n LOCATION:https://researchseminars.org/talk/BIMSA-ISS/1/ END:VEVENT END:VCALENDAR