Tue, September 1, 2020
Public Access Category: All |
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- Time:
- 5:30pm-6:30pm
- Description:
- Date and Time: Tuesday 1st September 2020, 5:30 pm IST - 6:30 pm IST
(joining time : 5:15 pm IST - 5:30 pm IST)
Google Meet link: https://meet.google.com/yqu-mvvy-jrs
Speaker: Matteo Varbaro, University of Genoa
Title: F-splittings of the polynomial ring and compatibly split
homogeneous ideals
Abstract: A polynomial ring R in n variables over a field K of positive
characteristic is F-split. It has many F-splittings. When K is a perfect
field every F-splitting is given by a polynomial g in R with the monomial
u^{p-1} in its support (where u is the product of all the variables)
occurring with coefficient 1, plus a further condition, which is not
needed if g is homogeneous (w.r.t. any positive grading). Fixed an
F-splitting s : R -> R, an ideal I of R such that s(I) is contained in I
is said compatibly split (w.r.t. the F-splittings). In this case R/I is
F-split. Furthermore, by Fedder’s criterion when I is a homogeneous ideal
of R, R/I is F-split if and only if I is compatibly split for some
F-splitting s : R -> R. If, moreover, u^{p-1} is the initial monomial of
the associated polynomial g of s w.r.t. some monomial order, then in(I) is
a square-free monomial ideal… In this talk I will survey these facts (some
of them classical, some not so classical), and make some examples,
focusing especially on determinantal ideals.
- Time:
- 7:00pm
- Description:
- The speaker is
Prof. Amritanshu Prasad from IMSc, Chennai. The following are the
details.
Title: Polynomials as Characters of Symmetric Groups.
Time: 7pm, Tuesday, September 1, 2020 (gate opens at 6:45pm).
Google meet link: meet.google.com/prm-feow-zwm.
Phone: (US) +1 740-239-3129 PIN: 706 683 026#
Abstract: Treating the variable $X_i$ as the number of $i$-cycles in a
permutation allows a polynomial in $X_1, X_2,\dotsc$ to be regarded as a
class function of the symmetric group $S_n$ for any positive integer $n$.
We present a simple formula for computing the average and signed average
of such a class function over the symmetric group. We use this formula to
investigate the dimension of $S_n$-invariant and $S_n$-sign-equivariant
vectors in polynomial representations of general linear groups.
This talk is based on joint work with Sridhar P Narayanan, Digjoy Paul,
and Shraddha Srivastava. Some of these results are available in the
preprint available at: http://arxiv.org/abs/2001.04112.