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SUMMARY:Digjoy Paul (IMSC Chennai)
DTSTART:20200526T160000Z
DTEND:20200526T170000Z
DTSTAMP:20260423T023943Z
UID:ADM-Davis/7
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/ADM-Davis/7/
 ">New approaches to the restriction problem</a>\nby Digjoy Paul (IMSC Chen
 nai) as part of UC Davis algebra & discrete math seminar\n\n\nAbstract\nGi
 ven an irreducible polynomial representation $W_n$ of the general linear g
 roup $GL_n$\, we can restrict it to the representations of the symmetric g
 roup $S_n$ that seats inside $GL_n$ as a subgroup. The restriction problem
  is to find a combinatorial interpretation of the restriction coefficient:
  the multiplicity of an irreducible $S_n$ modules in such restriction of $
 W_n$. This is an open problem (see OPAC 2021!) in algebraic combinatorics.
 \n\nIn Polynomial Induction and the Restriction Problem\, we construct the
  polynomial induction functor\, which is the right adjoint to the restrict
 ion functor from the category of polynomial representations of $GL_n$ to t
 he category of representations of $S_n$. This construction leads to a repr
 esentation-theoretic proof of Littlewood's Plethystic formula for the rest
 riction coefficient.\n\nCharacter polynomials have been used to study char
 acters of families of representations of symmetric groups (see Garsia and 
 Goupil )\, also used in the context of FI-modules by Church\, Ellenberg\, 
 and Farb (see FI-modules and stability for representations of symmetric gr
 oups).\n\nIn Character Polynomials and the Restriction Problem\, we comput
 e character polynomial for the family of restrictions of $W_n$ as $n$ vari
 es. We give an interpretation of the restriction coefficient as a moment o
 f a certain character polynomial. To characterize partitions for which the
  corresponding Weyl module has non zero $S_n$-invariant vectors is quite h
 ard. We solve this problem for partition with two rows\, two columns\, and
  for hook-partitions.\n\nThis is joint work with Sridhar Narayanan\, Amrit
 anshu Prasad\, and Shraddha Srivastava.\n
LOCATION:https://researchseminars.org/talk/ADM-Davis/7/
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