## Geometric Representation Theory

These seminars focus on research in representation theory and algebraic geometry.

http://www.math.toronto.edu/oyacobi/GRT

### Polynomial representation of general linear groups and categorifications of Fock space

by Oded Yacobi | University of Toronto
Time: 14:30 — 16:00  (Thursday, Jan. 20, 2011)
Location: BA6183, Bahen Center, 40 St. George St.
Abstract:
Consider a field $F$ of characteristic $p$. It is well known that the direct sum of the categories $Rep(S_k)$ categorifies the basic representation of the Kac-Moody algebra $g$, where $g$ is $\hat{sl}_p$ when $p>0$ and $sl_{\infty}$ when $p=0$. Moreover, the symmetric groups and general linear groups are related via Schur-Weyl duality. Therefore it is natural to ask for an analogue of the direct sum of categories $Rep(S_k)$ for the general linear groups. We will construct such a category as a limit category of polynomial representations of $GL_n$, and show that it categorifies the Fock space representation of $g$ (in the sense of Chuang and Rouquier). Time permitting, we will discuss some consequences of this categorification, such as how to recover the crystal of Fock space from this construction and how, from this perspective, Schur-Weyl duality categorifies the standard projection from Fock space onto the basic representation. This is based on joint work with Jiuzu Hong.

#### Dates in this series

· Friday, Feb. 05, 2010: Harmonic analysis on spherical varieties in terms of the wonderful compactification (Yiannis Sakellaridis)
· Friday, Feb. 12, 2010: String polytopes of Littelmann-Bernstein-Zelevinsky and Newton-Okounkov bodies (Kuimars Kaveh)
· Friday, Feb. 19, 2010: Tate's proof of the Residue Formula (Mikhail Mazin)
· Monday, Mar. 01, 2010: The 2-group of a linear auto-equivalences of an abelian category and its Lie 2-algebra (Xinwen Zhu)
· Friday, Mar. 05, 2010: Braid group actions in geometry (Sabin Cautis)
· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps ()
· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps ()
· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps ()
· Friday, Mar. 26, 2010: The hidden smoothness of the space of stable maps ()
· Tuesday, Mar. 30, 2010: Local duality and polarized Hodge modules (Christian Schnell)
· Friday, Apr. 09, 2010: Quantum cohomology of symplectic resolutions and derived auto-equivalences (Alexander Braverman)
· Tuesday, Sep. 21, 2010: Multiplicity Spaces in Classical Symplectic Branching (Oded Yacobi)
· Wednesday, Sep. 22, 2010: Modular tensor categories and modular functors: braided tensor categories and quantum groups (Joel Kamnitzer)
· Tuesday, Sep. 28, 2010: Cluster categories and higher-dimensional analogues (Hugh Thomas)
· Tuesday, Oct. 05, 2010: Categorification and geometric representation theory (Joel Kamnitzer)
· Tuesday, Oct. 12, 2010: Idempotents in $R$-trivial monoids (Chris Burg)
· Tuesday, Oct. 26, 2010: Webs and the Affine Grassmannian (Bruce Fontaine)
· Thursday, Nov. 04, 2010: Quantum insertion and Schur-Weyl duality (Sonya Berg)
· Thursday, Nov. 11, 2010: Geometrization of Principal Series Representation of $GL_N$ (Travis Schedler)
· Tuesday, Nov. 16, 2010: Symmetries of stability conditions (Parker Lowrey)
· Tuesday, Nov. 23, 2010: Categorified Heisenberg Actions on Hilbert Schemes (Sabin Cautis)
· Thursday, Jan. 20, 2011: Polynomial representation of general linear groups and categorifications of Fock space (Oded Yacobi)
· Tuesday, Jan. 25, 2011: The antigravity method in cyclotomic quotients of KLR algebras (Alex Hoffnung)
· Thursday, Jan. 27, 2011: Crystal models for the basic representation of the affine special linear Lie algebras (Chris Berg)
· Thursday, Feb. 03, 2011: Derived completion, derived descent and semi-infinite cohomology of quantum groups (Sergey Arkhipov)
· Tuesday, Feb. 08, 2011: Affine Mirkovic-Vilonen Polytopes (Joel Kamnitzer)
· Thursday, Feb. 24, 2011: Degenerations of Kazhdan-Lusztig varieties, and applications (Allen Knutson)
· Tuesday, Mar. 08, 2011: The derived category and the singularity category (Umut Isik)
· Tuesday, Mar. 15, 2011: The combinatorics of the affine Satake isomorphism (Alexander Braverman)
· Thursday, Mar. 24, 2011: 2-representation theory (Raphael Rouquier)
· Thursday, Apr. 07, 2011: Turaev--Viro invariants, Kitaev's model and extended TQFTs (Alexander Kirillov)
· Wednesday, May. 04, 2011: A brief introduction to the theory of descent,  (Omar Anatolin Camarena)
· Monday, Jul. 18, 2011: Gindikin-Karpelevich formulas and Plancherel Theorems (James Parkinson)
· Wednesday, Aug. 03, 2011: Quotient-closed subcategories of representations of quivers and sorting order on reflection groups (Hugh Thomas)
· Friday, Sep. 16, 2011: Equivariant coherent sheaves and categorifications of Affine Hecke algebras (Chris Dodd)
· Friday, Sep. 30, 2011: Quiver Representations in the Super-Category (Jaimie Thind)
· Friday, Oct. 07, 2011: Nilpotent semi-infinite homology of quantum groups and categorical completion (Sergey Akhipov)
· Friday, Oct. 21, 2011: The blob complex (Scott Morrison)
· Friday, Oct. 28, 2011: Equivariant map algebras: irreducible representations and block decompositions (Alistair Savage)
· Friday, Nov. 04, 2011: Graded Expansions of Groups (Peter Lee)
· Friday, Nov. 04, 2011: `Quadraticity' of Groups (Peter Lee)
· Friday, Nov. 11, 2011: Spin Networks (Roland van der Veen)
· Friday, Nov. 18, 2011: On the structure of intrinsic topological order in 2D (Oliver Buerschaper)
· Friday, Nov. 25, 2011: Jacobi Factors of Quasihomogeneous Plane Curve Singularities (Mikhail Mazin)
· Friday, Dec. 02, 2011: Promotion Operators and Crystal Bases (Tom Denton)
· Tuesday, Dec. 13, 2011: Quantum affine algebras, Braid groups, and Heisenberg categorification (Tony Licata)
· Friday, Jan. 13, 2012: Bruhat cells and atlases thereof (Allen Knutson)
· Friday, Jan. 20, 2012: The classification of irreducible mod p representations of a p-adic GL_n (Florian Herzig)
· Friday, Jan. 27, 2012: Categorifications of the characteristic polynomial of hyperplane arrangements (Zsuzsanna Dancso)
· Friday, Mar. 02, 2012: L-functions and their generating functions (Avner Segal)
· Monday, Mar. 12, 2012: Laminations, Affine Buildings, and Invariants of Tensor Products of Representations (Ian Le)
· Friday, Mar. 23, 2012: Combinatorics of twisted Higgs bundles: g=0 and beyond (Steven Rayan)
· Monday, Mar. 26, 2012: Real variations of stability conditions and representation theory (Roman Bezrukavnikov)
· Thursday, Apr. 05, 2012: Coherent DG-modules over the de Rham DG-algebra (Sergey Rybakov)
· Thursday, Sep. 27, 2012: Classifying subfactors (Dave Penneys)
· Thursday, Oct. 04, 2012: Introduction to Geometric Complexity Theory (Josh Grochow)
· Thursday, Oct. 11, 2012: Crystals, geometric Satake, and non-negative tropical flag variety (Joel Kamnitzer)
· Friday, Oct. 19, 2012: Ramified Satake Isomorphisms (Masoud Kamgarpour)
· Friday, Oct. 26, 2012: Combinatorial rules for branching to symmetric subgroups (Allen Knutson)
· Thursday, Nov. 01, 2012: Crystals, geometric Satake, and the non-negative tropical flag variety II (Joel Kamnitzer)
· Thursday, Nov. 08, 2012: Crystals, geometric Satake, and non-negative tropical flag variety III (Joel Kamnitzer)
· Thursday, Nov. 08, 2012: Character varieties as completely integrable systems (Adam Sikora)
· Thursday, Nov. 22, 2012: Crystals, geometric Satake, and non-negative tropical flag variety III (Joel Kamnitzer)
· Thursday, Nov. 29, 2012: Crystals, geometric Satake, and the non-negative tropical flag variety IV (Joel Kamnitzer)
· Friday, Nov. 30, 2012: The colored HOMFLY polynomial is q-holonomic (Stavros Garoufalidis)
· Thursday, Jan. 17, 2013: Twisted Geometric Satake Equivalence (Bhairav Singh)
· Thursday, Jan. 24, 2013: Extended conformal field theories (Andre Henriques)
· Thursday, Feb. 07, 2013: Quantum cohomology of hypertoric spaces, Bethe equations and mirror formulae (Michael McBreen)