Research Summary
نویسنده
چکیده
These notes are based on my Advancement to Candidacy Lecture Notes, given on Friday, May 27, 2016. I expect I will be expanding them in due time. My aim is to tackle some questions relating to the modular representation theory of symmetric groups. Sections 1 through 3 give some of the established results on the subject. Section 1 describes the CDE triangle associated to the modular representation theory, and expands it with the Hecke algebra at a root of unity serving as an intermediate between characteristic 0 and characteristic p. Section 2 describes the results that were conjectured and proved in the 1990s. Section 3 explains why these theorems don’t give us the desired decomposition matrix. Section 4 describes what the recently discovered Cyclotomic Quiver Hecke Algebras have to offer. Section 5 covers the questions that I hope to help address. Section 6 gives the integral version of the main theorem of section 2. Let Sn denote the symmetric group on n letters. The main objects of interest will be kSn, for k a field; in particular, the finite fields k = Fp. For k a field and A a k-algebra, let A-mod denote the category of finite-dimensional A-modules, and let A-proj denote the full sub-category of finite-dimensional projective A-modules.
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