Tue, September 1, 2020
Public Access

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September 2020
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5:00pm [5:30pm] Matteo Varbaro, University of Genoa
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.

7:00pm [7:00pm] Amritanshu Prasad: IMSc, Chennai
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.