3:35 pm Tuesday, September 27, 2016
Junior Geometry Seminar: Mirror symmetry for log Calabi-Yaus and combinatorial representation theory by Tim Magee in RLM 12.166
Roughly speaking, a log Calabi-Yau is a space that comes equipped with a volume form in a natural way. Let 𝑋 be an affine log CY. We can partially compactify 𝑋 by adding divisors along which Ω has a pole. The set of these divisors says a lot about 𝑋's geometry — for a torus (the simplest example of a log CY) this set is just the cocharacter lattice. We can actually give this set a geometrically motivated multiplication rule too, which I hope not to get into. But this multiplication rule allows us to construct an algebra 𝐴, defined purely in terms of the geometry of 𝑋, and conjecturally 𝐴 is the algebra of regular functions on the mirror to 𝑋. Viewed as a vector space, 𝐴 naturally comes with a basis — the divisors we used to define it. So we get a canonical basis for the space of regular functions on 𝑋's mirror. Many objects of interest to representation theorists (semi-simple groups, flag varieties, Grassmannians...) are nice partial compactifications of log CY's. This gives us a chance to use the machinery of log CY mirror symmetry to get results in rep theory. Maybe the most obvious application given what I've said so far is finding a canonical basis for irreducible representations of a group. I'll discuss this, as well as how this machinery reproduces cones that make combinatorial rep theorists feel warm and fuzzy inside (like the Gelfand-Tsetlin cone and the Knutson-Tao hive cone). All of the technology involved is a souped-up version of something from the world of toric varieties. I'll discuss things in this context as much as possible in an attempt to keep everything accessible. Submitted by
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