BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Anne Schilling (UC Davis) DTSTART:20200415T040000Z DTEND:20200415T053000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/1 DESCRIPTION:Title: Crystal for stable Grothendieck polynomials\, part 2\nby Anne Sch illing (UC Davis) as part of IMSc algebraic combinatorics seminar\n\n\nAbs tract\nThis is part 2 of a two-part seminar.\n\nWe introduce a new crystal on decreasing factorizations on 321-avoiding elements in the 0-Hecke mono id which we call ⋆-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the ⋆-crysta l intertwines with the crystal on set-valued tableaux recently introduced by Monical\, Pechenik and Scrimshaw. We also define a new insertion from d ecreasing factorization in the 0-Hecke monoid to pairs of (transposes of) semistandard Young tableaux and prove several properties about this new in sertion\, in particular its relation to the Hecke insertion and the uncrow ding algorithm. The new insertion also intertwines with the crystal operat ors.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Arvind Ayyer (IISc\, Bangalore) DTSTART:20200422T093000Z DTEND:20200422T110000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/2 DESCRIPTION:Title: Combinatorics of an exclusion process driven by an asymmetric tracer< /a>\nby Arvind Ayyer (IISc\, Bangalore) as part of IMSc algebraic combinat orics seminar\n\n\nAbstract\nWe consider an exclusion process on a periodi c one-dimensional lattice where all particles perform simple symmetric exc lusion except for a _tracer particle_\, which performs partially asymmetri c exclusion with forward and backward rates p and q respectively. This pro cess and its variants have been investigated starting with Ferrari\, Golds tein and Lebowitz (1985) motivated by questions in statistical physics. We prove product formulas for stationary weights and exact formulas for the nonequilibrium partition function in terms of combinatorics of set partiti ons. We will also compute the current\, and the density profile as seen by the test particle. Time permitting\, we will illustrate the ideas involve d in performing asymptotic analysis. This talk is based on the preprint a rXiv:2001.02425.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/2/ END:VEVENT BEGIN:VEVENT SUMMARY:G. Arunkumar (IISER Mohali) DTSTART:20200429T053000Z DTEND:20200429T070000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/3 DESCRIPTION:Title: Chromatic Symmetric Function of Graphs from Borcherds Lie Algebra \nby G. Arunkumar (IISER Mohali) as part of IMSc algebraic combinatorics s eminar\n\n\nAbstract\nLet $\\mathfrak g$ be a Borcherds algebra with the a ssociated graph $G$. We prove tha\\\nt the chromatic symmetric function of $G$ can be recovered from the Weyl denominators of $\\mathfrak g$ and thi s gives a Lie theoretic proof of Stanley’s expression for chromatic symm etric function in terms of power sum symmetric functions. Also\, this give s an expression for the chromatic symmetric function of $G$ in terms of ro ot multiplicities of $\\mathfrak g$. We prove a modified Weyl denominator identity for Borcherds algebras which is an extension of the celebrated cl assical Weyl denominator identity and this plays an important role in the proof our results. The absolute value of the linear coefficient of the chr omatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our main theorem\, we prove that certain coefficients a ppearing in the above said expression of chromatic symmetric function is e qual to the chromatic discriminant of $G$. Also\, we find a connection bet ween the Weyl denominators and the $G$-elementary symmetric functions. Usi ng this connection\, we give a Lie-theoretic proof of non-negativity of co efficients of $G$-power sum symmetric functions.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Mike Zabrocki (York University) DTSTART:20200506T150000Z DTEND:20200506T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/4 DESCRIPTION:Title: Symmetric group characters as symmetric functions\nby Mike Zabroc ki (York University) as part of IMSc algebraic combinatorics seminar\n\n\n Abstract\nI will present a basis of the symmetric functions whose evaluati ons are irreducible\ncharacters of the symmetric group in the same way tha t the evaluations of Schur\nfunctions are irreducible characters of the ge neral linear group.  These symmetric\nfunctions are related to character polynomials (that go back to a paper of\nFrobenius in 1904) but they have the advantage that we are able to use the Hopf\nstructure of the symmetric functions to compute with them.  In addition\, they\nindicate that the c ombinatorics of Kronecker coefficients is governed by multiset\ntableaux.   We use this basis to give a combinatorial interpretation for the tensor \nproducts of the form\n$$\\chi^{(n-|\\lambda|\,\\lambda)} \\otimes \\chi ^{(n-a_1\,a_1)} \n\\otimes \\chi^{(n-a_2\,a_2)} \\otimes \\cdots \\oti mes \\chi^{(n-a_r\,a_r)}$$\nwhere $\\lambda$ is a partition and $a_1\, a_ 2\, \\ldots\, a_r$ are non-negative\nintegers.\n\nThis is joint work with Rosa Orellana.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Nate Harman (The Univesity of Chicago) DTSTART:20200513T150000Z DTEND:20200513T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/5 DESCRIPTION:Title: Intermediate Algebraic Structure in the Restriction Problem\nby N ate Harman (The Univesity of Chicago) as part of IMSc algebraic combinator ics seminar\n\n\nAbstract\nThe restriction problem refers to understanding in a combinatorial sense the\ndecomposition of an irreducible representat ion of GL_n as a representation of S_n. In\nthis talk\, I will discuss so me of the intermediate algebraic structures that arise when\nstudying this problem which constrain the symmetric group representations that appear a nd\n(hopefully) give some insight into the general problem. Things I will mention include: \nRepresentation stability\, the rook monoid\, the group of monomial matrices\, and a certain\nsubalgebra of the universal envelop ing algebra which seems to have interesting\ncombinatorial properties.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Amritanshu Prasad (The Institute of Mathematical Sciences\, Chenna i) DTSTART:20200521T150000Z DTEND:20200521T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/6 DESCRIPTION:Title: A timed version of the plactic monoid\nby Amritanshu Prasad (The Institute of Mathematical Sciences\, Chennai) as part of IMSc algebraic co mbinatorics seminar\n\n\nAbstract\nLascoux and Schutzenberger introduced t he plactic monoid as a tool to prove the Littlewood-Richardson rule. The p lactic monoid is the quotient of the free monoid on an ordered alphabet mo dulo Knuth relations. In this talk I will explain how their theory can be generalized to timed words\, which are words where each letter occurs for a positive amount of time rather than discretely. This generalization give s an organic approach to piecewise-linear interpolations of correspondence s involving semi-standard Young tableaux. This talk is based on the arxiv preprint arXiv:1806.04393.\n\nZoom Meeting ID: 816 4919 7982.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Deniz Kus (Ruhr-University Bochum) DTSTART:20200611T150000Z DTEND:20200611T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/7 DESCRIPTION:Title: Quiver varieties and their combinatorial crystal structure\nby De niz Kus (Ruhr-University Bochum) as part of IMSc algebraic combinatorics s eminar\n\n\nAbstract\nThe aim of this talk is to describe combinatorially the crystal operators on the geometric realization of crystal bases in ter ms of irreducible components of quiver varieties. As a consequence of this description one can extend the geometric description to an affine crystal isomorphic to a Kirillov-Reshetikhin crystal. The underlying combinatoric s is in terms of Auslander Reiten quivers.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Inna Entova (Ben Gurion University) DTSTART:20200625T150000Z DTEND:20200625T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/8 DESCRIPTION:Title: Deligne categories and stable Kronecker coefficients\nby Inna Ent ova (Ben Gurion University) as part of IMSc algebraic combinatorics semina r\n\n\nAbstract\nIn this talk\, I will present an application of the theor y of Deligne categories to the study of Kronecker coefficients.\n\n < p>Kronecker coefficients are structural constants for the category $\\math rm{Rep}(S_n)$ of finite-dimensional representations of the symmetric g\\\n roup\; namely\, given three irreducible representations $\\mu\, \\tau\, \\ lambda$ of $S_n$\, the Kronecker coefficient $\\mathrm{Kron}( \\lambda\, \ \mu\, \\tau)$ i\\\ns the multiplicity of $\\lambda$ inside $\\mu \\otimes \\tau$.\n The study of Kronecker coefficients has been described a s "one of the main problems in the combinatorial representation theory of the symmetri\\\nc group"\, yet very little is known about them.

\n\n

I will define a "stable" version of the Kronecker coefficients (due to Murnaghan)\, which generalizes\n both Kronecker coefficientsand Li ttlewood-Richardson coefficients (structural constants for general linear\ n groups).

\n\n

It turns out that the stable Kronecker c oefficients appear naturally as structural constants in the\n Deligne categories $\\mathrm{Rep}(S_t)$\, which are interpolations of the categor ies $\\mathrm{Rep}(S_n)$ to complex $t$. I\n will explain this phenom enon\, and show that the categorical properties of $\\mathrm{Rep}(S_t)$ al low us not only to\n recover known properties of the stable Kroneck er coefficients\, but also obtain new identities.

\n

\n This is a report on my project from 2014.

\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeanne Scott (Universidad de los Andes) DTSTART:20200604T150000Z DTEND:20200604T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/9 DESCRIPTION:Title: What's the right notion of content for the Young-Fibonacci lattice?\nby Jeanne Scott (Universidad de los Andes) as part of IMSc algebraic c ombinatorics seminar\n\n\nAbstract\nThe Young-Fibonacci lattice $\\Bbb{YF} $ is a ranked lattice invented by R. Stanley as an example of a differenti al poset\; a nice consequence of this feature is that saturated chains (wh ich a fixed top) are counted by a generalized hook-length formula. In 1994 S. Okada showed that $\\Bbb{YF}$ is also the branching poset for a tower of complex semi-simple algebras $\\frak{F}(n)$\, each having a simple Coxe ter-like presentation. The representation theory of these algebras strongl y parallels the story of the symmetric groups $S(n)$ --- in particular eac h element w of rank $\\mathrm{rk}(w) = n$ in the $\\Bbb{YF}$ lattice corre sponds to an irreducible representation $V(w)$ of $\\frak{F}(n)$ whose bas is is indexed by saturated chains in the $\\Bbb{YF}$ lattice ending at $w$ . Furthermore there is a theory of $\\Bbb{YF}$-Schur functions obeying a L ittlewood-Richardson rule whose structure coefficients coincide with the i nduction product multiplicities for representations of the Okada algebras .\n\nAs in any tower of semi-simple algebras with a simple braching poset\ , we may define the Gelfand-Tsetlin algebra $\\mathrm{GT}(n)$ as the (maxi mal) commutative subalgebra of $\\frak{F}(n)$ generated by the centers $Z\ \frak{F}(1)\, Z\\frak{F}(2)\, \\dots\, Z\\frak{F}(n)$. The problem I would like to address is how to find (additive) Jucys-Murphy elements\, namely an infinite sequence of elements $J(n)$ such that:\n\n
\n (1) each $J(n )$ resides in $\\mathrm{GT}(n)$
\n\n (2) $J(1)\, \\dots\, J(n)$ generat e $\\mathrm{GT}(n)$
\n\n (3) the sum $J(1) + \\cdots + J(n)$ resides in $Z\\frak{F}(n)$
\n\n (4) each $J(k)$ acts diagonally on the irreducibl e representation $V(w)$ and its eigenvalue\, with respect to a basis vecto r indexed by a saturated chain $u(0) \\lhd \\cdots \\lhd u(n)$\, depend on ly on the covering relation $u(k-1) \\lhd u(k)$ in $\\Bbb{YF}$.
\n\n\nT his local eigenvalue $c(u \\lhd v)$ is called the content of covering rela tion $u \\lhd v$ with respect to the choice of Jucys-Murphy generators. Ke ep in mind that there are many different systems of elements $J(n)$ satisf ying properties (1)\, (2)\, (3)\, and (4). However\, not any assignment of covering weights $c(u \\lhd v)$ can be realized as contents for such a sy stem. Indeed a necessary condition requires that two saturated chains coin cide if and only if the corresponding sequences of covering weights are eq ual\; see recent work of S. Doty et. al.\n\nSince the Jucys-Murphy problem is under-determined it is natural to use the tower of symmetric groups $S (n)$ together with its branching poset\, the Young lattice $\\Bbb{Y}$\, as a guide to impose further constraints. For example\, one might try determ ine a system of Jucys-Murphy elements by forcing the attending system of c ontents to satisfy a specialization formula for the $\\Bbb{YF}$-Schur func tions in analogy with the principal specialization of classical Schur func tions. This is work in progress.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Shubham Sinha (UCSD) DTSTART:20200618T150000Z DTEND:20200618T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/10 DESCRIPTION:Title: Random $t$-cores and hook lengths in random partitions\nby Shubh am Sinha (UCSD) as part of IMSc algebraic combinatorics seminar\n\n\nAbstr act\nFix $t \\geq 2$. We first give an asymptotic formula for certain sums of the number of $t$-cores. We then use this result to compute the distri bution of the size\nof the $t$-core of a uniformly random partition of an integer $n$. We show that this converges weakly to a gamma distribution af ter appropriate rescaling. As a consequence\, we find that the size of the $t$-core is of the order of $\\sqrt{n}$ in expectation. We then apply thi s result to show that the probability that $t$ divides the hook length of a uniformly random cell in a uniformly random partition equals $1/t$ in th e limit. Finally\, we extend this result to all modulo classes of $t$ usin g abacus representations for cores and quotients. This talk is based on th e arxiv preprint arXiv:1911. 03135.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Sheila Sundaram (Pierrepont School\, Westport\, CT\, USA) DTSTART:20200702T150000Z DTEND:20200702T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/11 DESCRIPTION:Title: Plethystic inversion and representations of the symmetric group\ nby Sheila Sundaram (Pierrepont School\, Westport\, CT\, USA) as part of I MSc algebraic combinatorics seminar\n\n\nAbstract\nIn this talk we will su rvey the many instances of plethystic inversion that occur in the represen tation theory of the symmetric group $S_n$. Perhaps the first such formula is due to Cadogan. The\nLie representation of $S_n\,$ arising from the fr ee Lie algebra\, appears here. We will discuss the equivalence\n of Cadoga n's formula to Thrall's decomposition of the regular representation\, and to many other phenomena in\n a wide variety of contexts. New decomposition s of the regular representation will be presented. Some of this\n material appears in the following papers:\n\n arXiv:1803.09368\n\n arXiv:2003.10700.\n\n arXiv:2006.01896\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Digjoy Paul (IMSc\, Chennai) DTSTART:20200716T150000Z DTEND:20200716T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/12 DESCRIPTION:Title: Quasi p-Steinberg Character for Symmetric\, Alternating Groups and t heir Double Covers\nby Digjoy Paul (IMSc\, Chennai) as part of IMSc al gebraic combinatorics seminar\n\n\nAbstract\nGiven a finite group of Lie t ype in characteristic $p$\,  Steinberg constructed a distinguished ordin ary representation of dimension equals to the\nthe cardinality of a Sylo w-$p$-subgroup and whose character\, which is now known as $p$-Steinberg character\, vanishes except at $p$-regular elements. \n \nThe followin g question was raised by W. Feit\, and was answered by M. R. Darafsheh fo r the alternating group or the projective special linear group:\n "Let $G $ be a finite simple group of order divisible by the prime $p$\, and suppo se that $G$ has a $p$-Steinberg character. Does it follow that $G$ is a\ns emisimple group of Lie type in characteristic $p$?"\n\nThis motivates us t o define Quasi $p$-Steinberg character for finite groups.\nAn irreducib le character of a finite group $G$ is called quasi $p$-Steinberg\n for a prime $p||G|$ if it is non zero on every $p$-regular element of $G$. \nIn this talk\, we discuss the existence of quasi $p$-Steinberg Characters of Symmetric as well as  Alternating groups and their double covers. On the \nway\, we also answer a question\, similar to Feit\, asked by  Dipendra Prasad.\nThis talk is based on ongoing work with Pooja Singla.\n\nReferen ces:\n\n1. J. E. Humphreys\, The Steinberg representation\,1987.\n\n2. W . Feit\, Extending Steinberg Characters\,1993.\n\n3. M. R. Darafsheh\, $p $-Steinberg Characters of Alternating and Projective Special Linear Groups 1995.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Digjoy Paul (IMSc\, Chennai) DTSTART:20200806T150000Z DTEND:20200806T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/13 DESCRIPTION:Title: Quasi $p$-Steinberg Characters of double covers of Symmetric and Alt ernating groups\nby Digjoy Paul (IMSc\, Chennai) as part of IMSc algeb raic combinatorics seminar\n\n\nAbstract\nAn irreducible character of a fi nite group $G$ is called Quasi $p$-Steinberg for a prime $p$ if it takes n on-zero value on every $p$-regular element of $G$.\n\nIn this talk\, we s hall recall some combinatorial aspects of the representation theory of dou ble covers of Symmetric and Alternating groups. Then we discuss the existe nce of Quasi\n$p$-Steinberg Characters of those groups. This talk is based on ongoing work with Pooja Singla.\n\n\nSuggested readings:\n\n1. A. O. M orris\, The spin representation of the symmetric group\, Proc. London Math . Soc. (3)\, 12 (1962).\n\n2. J. R. Stembridge\, Shifted tableaux and the projective representations of the symmetric groups. Adv. in Math. 74 (1989 ).\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Ryan Vinroot (William and Mary) DTSTART:20200813T150000Z DTEND:20200813T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/14 DESCRIPTION:Title: Generating Functions for Involutions and Character Degree Sums in Fi nite Groups of Lie Type\nby Ryan Vinroot (William and Mary) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nGiven a finite group $G$\, it is a result of Frobenius and Schur that all\ncomplex irreducible representations of $G$ may be defined over the reals if and only if\nthe c haracter degree sum of $G$ is equal to the number of involutions of $G$.  We use\nthis result and generatingfunctionology to study the real represe ntations of finite\ngroups of Lie type\, and to obtain some new combinator ial identities.  We will begin with\nexamples of Weyl groups\, then discu ss joint work with Jason Fulman on finite general\nlinear and unitary grou ps\, and then give more recent results for finite symplectic and\northogon al groups.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Anupam Kumar Singh (IISER Pune) DTSTART:20200820T043000Z DTEND:20200820T053000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/15 DESCRIPTION:Title: Asymptotics of powers in finite reductive groups\nby Anupam Kuma r Singh (IISER Pune) as part of IMSc algebraic combinatorics seminar\n\n\n Abstract\nLet $G$ be a connected reductive group defined over a finite fie ld $\\mathbf F_q$. Fix an integer $M >1$\, and consider the power map $x$ going to $x^M$ on G. We denote the image of $G(\\mathbf F_q)$ under this m ap by $G(\\mathbf F_q)^M$ and estimate what proportion of regular semisimp le\, semisimple and regular elements of $G(\\mathbf F_q)$ it contains. We prove that as q tends to infinity\, all of these proportions are equal and provide a formula for the same. We also calculate this more explicitly fo r the groups $GL(n\, q)$ and $U(n\, q)$.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Sami H Assaf (University of Southern California) DTSTART:20200903T033000Z DTEND:20200903T043000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/16 DESCRIPTION:Title: Pieri rules for polynomials\nby Sami H Assaf (University of Sout hern California) as part of IMSc algebraic combinatorics seminar\n\n\nAbst ract\nSchur functions are an amazing basis of symmetric functions original ly defined as characters of irreducible modules for of $GL_n$. The Pieri r ule for the product of a Schur function and a single row Schur function is a multiplicity-free branching rule with a beautiful combinatorial interpr etation in terms of adding boxes to a Young diagram. Key polynomials are a n interesting basis of the polynomial ring originally defined as character s of submodules for irreducible $GL_n$ modules under the action of upper t riangular matrices. In joint work with Danjoseph Quijada\, we give a Pieri rule for the product of a key polynomial and a single row key polynomial. While this formula has signs\, it is multiplicity-free and has an interpr etation in terms of adding balls to a key diagram\, perhaps after dropping some balls down. Time permitting\, I’ll give applications to Schubert p olynomials where the signs cancel to give a positive Pieri formula.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Subhajit Ghosh (Indian Institute of Science) DTSTART:20200827T053000Z DTEND:20200827T063000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/17 DESCRIPTION:Title: Total variation cutoff for random walks on some finite groups\nb y Subhajit Ghosh (Indian Institute of Science) as part of IMSc algebraic c ombinatorics seminar\n\n\nAbstract\nThis presentation will be on the mixin g times for three random walk models. Specifically these are the random wa lks on the alternating group\, the group of signed permutations and the co mplete monomial group. The details for the models are given below:
\n The random walk on the alternating group: We investigate the pr operties of a random walk on the alternating group $A_n$ generated by $3$- cyclesof the form $(i\,n-1\,n)$ and $(i\,n\,n-1)$. We call this the tra nspose top-$2$ with random shuffle. We find the spectrum of the transi tion matrixof this shuffle. We obtain the sharp mixing time by proving the total variation cutoff phenomenon at $\\left(n-\\frac{3}{2}\\right)\\log n$ for this shuffle.
\nThe random walk on the group of signed permut ations: We consider a random walk on the hyperoctahedral group $B_n$ g enerated by the signed permutations of the form $(i\,n)$ and $(-i\,n)$ for $1\\leq i\\leq n$. We call this the flip-transpose top with random shu ffle on $B_n$. We find the spectrum of the transition probability matr ix for this shuffle. We prove that this shuffle exhibits the total variati on cutoff phenomenon with cutoff time $n\\log n$. Furthermore\, we show th at a similar random walk on the demihyperoctahedral group $D_n$ generated by the identity signed permutation and the signed permutations of the form $(i\,n)$ and $(-i\,n)$ for $1\\leq i< n$ also has a cutoff at $\\left(n-\ \frac{1}{2}\\right)\\log n$.
\nThe random walk on the complete monom ial group: Let $G_1\\subseteq\\cdots\\subseteq G_n \\subseteq\\cdots $ be a sequence of finite groups with $|G_1|>2$. We study the properties of a random walk on the complete monomial group $G_n\\wr S_n$ generated by t he elements of the form $(\\text{e}\,\\dots\,\\text{e}\,g\;\\text{id})$ an d $(\\text{e}\,\\dots\,\\text{e}\,g^{-1}\,\\text{e}\,\\dots\,\\text{e}\,g\ ;(i\,n))$ for $g\\in G_n\,\\\;1\\leq i< n$. We call this the warp-trans pose top with random shuffle on $G_n\\wr S_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mix ingtime for this shuffle is of order $n\\log n+\\frac{1}{2}n\\log (|G_n|-1 )$. We also show that this shuffle satisfies cutoff phenomenon with cutoff time $n\\log n$ if $|G_n|=o(n^{\\delta})$ for all $\\delta>0$.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Krishnan Rajkumar (Jawaharlal Nehru University) DTSTART:20200910T083000Z DTEND:20200910T093000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/18 DESCRIPTION:Title: Generalizations of the Selberg integral and combinatorial connection s\nby Krishnan Rajkumar (Jawaharlal Nehru University) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nWe'll briefly recall the hi story of the Selberg Integral and several variants. We'll also go through the proof of some of them like Aomoto's integral before focusing on known and possibly new integrals involving Schur polynomials and Jack polynomial s. We shall note the implications that these integrals seem to count (afte r a suitable normalization) the number of standard young tableaux of skew shapes\, before conjecturing the existence of several Naruse-type hook len gth formulas. Finally we will explain how these integrals arise in number theoretic problems.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Sumana Hauti (IISc\, Bangalore) DTSTART:20200917T083000Z DTEND:20200917T100000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/19 DESCRIPTION:Title: On Schur multipliers and projective representations of Heisenberg gr oups\nby Sumana Hauti (IISc\, Bangalore) as part of IMSc algebraic com binatorics seminar\n\n\nAbstract\nThe study of projective representations has a long history starting with the pioneering work of Schur for finite g roups which involves understanding homomorphisms from a group into the pro jective linear groups. In this study\, an important role is played by a gr oup called the Schur multiplier. In this talk\, we shall describe the Schu r multiplier of the finite as well as infinite discrete Heisenberg groups and their t-variants. We shall discuss the representation groups of these Heisenberg groups and through these give a construction of their finite-di mensional complex projective irreducible representations. This is joint wo rk with Pooja Singla.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Rijubrata Kundu (IISER Pune) DTSTART:20200924T150000Z DTEND:20200924T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/20 DESCRIPTION:by Rijubrata Kundu (IISER Pune) as part of IMSc algebraic comb inatorics seminar\n\n\nAbstract\nLet $M\\geq 2$ be any integer. Consider t he set\n$\\text{GL}(n\,q)^M=\\{x^M|x\\in \\text{GL}(n\,q)\\}$\, which\nis the set of all $M^{th}$ powers in the group $\\text{GL}(n\,q)$. In this\nt alk\, we will obtain generating functions for\n(a) the proportion of regul ar and regular semsimple elements in\n$\\text{GL}(n\,q)^M$\, assuming $(M\ ,q)=1$\,\n (b) the proportion of semisimple and all elements which are $M ^{th}$ powers\nwhen $(M\,q)=1$\, and $M$ is a power of a prime.\nTime perm itting we will also discuss the other extreme\, where we assume $M$\nis a prime and $q$ is a power of $M$.\nThis is a joint work with Dr. Anupam Sin gh.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/20/ END:VEVENT BEGIN:VEVENT SUMMARY:S Viswanath (The Institute of Mathematical Sciences\, Chennai) DTSTART:20201001T083000Z DTEND:20201001T093000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/21 DESCRIPTION:Title: Saturation for refined Littlewood-Richardson coefficients-I\nby S Viswanath (The Institute of Mathematical Sciences\, Chennai) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nThe Littlewood-Richard son (LR) coefficients are the multiplicities of irreducible representation s occurring in the tensor product of two irreducible polynomial representa tions of GL_n. To each permutation 'w' in S_n\, we associate a 'w-refineme nt' of the LR coefficients. These correspond to multiplicities in the so-c alled Kostant-Kumar submodules of the tensor product\, or equivalently of multiplicities in "excellent filtrations" of Demazure modules. We prove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231 -avoiding\, by adapting the proof via hives of the classical saturation co njecture due to Knutson-Tao. This is a report of work-in-progress with Mri gendra Singh Kushwaha and KN Raghavan. This talk will span two seminar day s (Oct 1 and 8). In the first part\, we describe the setting of the proble m and the result. In the second part\, we recall the key steps in the Knut son-Tao proof of the saturation conjecture via hives and indicate how it c an be adapted to our case.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/21/ END:VEVENT BEGIN:VEVENT SUMMARY:S Viswanath (The Institute of Mathematical Sciences\, Chennai) DTSTART:20201015T053000Z DTEND:20201015T063000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/22 DESCRIPTION:Title: Saturation for refined Littlewood-Richardson coefficients-2\nby S Viswanath (The Institute of Mathematical Sciences\, Chennai) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nThe Littlewood-Richard son (LR) coefficients are the multiplicities of irreducible representation s occurring in the tensor product of two irreducible polynomial representa tions of GL_n. To each permutation 'w' in S_n\, we associate a 'w-refineme nt' of the LR coefficients. These correspond to multiplicities in the so-c alled Kostant-Kumar submodules of the tensor product\, or equivalently of multiplicities in "excellent filtrations" of Demazure modules. We prove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231 -avoiding\, by adapting the proof via hives of the classical saturation co njecture due to Knutson-Tao. This is a report of work-in-progress with Mri gendra Singh Kushwaha and KN Raghavan. This talk will span two seminar day s (Oct 1 and 8). In the first part\, we describe the setting of the proble m and the result. In the second part\, we recall the key steps in the Knut son-Tao proof of the saturation conjecture via hives and indicate how it c an be adapted to our case.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science) DTSTART:20201112T083000Z DTEND:20201112T093000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/23 DESCRIPTION:Title: Polya frequency sequences: analysis meets algebra\nby Apoorva Kh are (Indian Institute of Science) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nI will provide an introduction to the theory of Pol ya frequency (PF) sequences. The first half includes examples (including l og-concave/unimodal sequences\, Hilbert series) and classical results on g enerating functions of PF sequences (with some proofs\, and one related Hy pothesis). In the second half\, I discuss connections from total positivit y to old and new phenomena involving symmetric functions. (Partly joint wi th Terence Tao.)\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Ajmain Yamin (Stony Brook) DTSTART:20201022T150000Z DTEND:20201022T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/24 DESCRIPTION:Title: Filtering Grassmannian Cohomology via k-Schur Functions\nby Ajma in Yamin (Stony Brook) as part of IMSc algebraic combinatorics seminar\n\n \nAbstract\nThis talk concerns the cohomology rings of complex Grassmannia ns. In 2003\, Reiner and Tudose conjectured the form of the Hilbert series for certain subalgebras of these cohomology rings. We build on their work in two ways. First\, we conjecture two natural bases for these subalgebra s that would imply their conjecture using notions from the theory of k-Sch ur functions. Second\, we formulate an analogous conjecture for Lagrangian Grassmannians.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Rosa Orellana (Dartmouth) DTSTART:20201029T150000Z DTEND:20201029T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/25 DESCRIPTION:Title: The Howe duality for the symmetric group\nby Rosa Orellana (Dart mouth) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nClas sical Howe dualities provide a representation theoretical framework for\nc lassical invariant theory.  In the classical Howe duality\,\n$GL_n(\\math bb{C})$ is dual to $GL_k(\\mathbb{C})$ when acting on the\npolynomial ring in the variables $x_{i\,j}$ where $1\\leq i\\leq n$ and $1\\leq\nj\\leq k $.  In this talk\,  I will introduce a multiset partition algebra\,\n$MP _k(n)$\, as the Howe dual to the symmetric group $S_n$.  \n\nThis is join t work with Mike Zabrocki.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Amrutha P (IISER Thiruvananthapuram) DTSTART:20201105T150000Z DTEND:20201105T163000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/26 DESCRIPTION:Title: On the determinant of representations of generalized symmetric group s $Z_r\\wr S_n$.\nby Amrutha P (IISER Thiruvananthapuram) as part of I MSc algebraic combinatorics seminar\n\n\nAbstract\nThe problem of enumerat ion of the number of irreducible representations of the symmetric group wi th a non trivial determinant was first considered by L. Solomon and later posed by Stanley in his book. Recently\, several authors have characterize d and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results\, we are int erested in the study of the determinant of irreducible representations of the generalized symmetric groups\, $Z_r \\wr S_n$. We give an explicit for mula to compute the determinant of an irreducible representation of $Z_r \ \wr S_n$. Also\, for a given integer $n$\, and a prime number $r$ and $\\z eta$ a nontrivial multiplicative character of $Z_r \\wr S_n$ with $n\\lt r $\, we obtain an explicit formula to compute $N_\\zeta(n)$\, the number of irreducible representations of $Z_r \\wr S_n$ whose determinant is $\\zet a$.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Krishnan Rajkumar (JNU) DTSTART:20201119T083000Z DTEND:20201119T093000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/27 DESCRIPTION:Title: Selberg integrals involving Schur polynomials and combinatorial conn ections\nby Krishnan Rajkumar (JNU) as part of IMSc algebraic combinat orics seminar\n\n\nAbstract\nIn the second take on this topic\, we will sk ip the history and focus on the combinatorial connections. In the first pa rt of the talk we will (re)derive Selberg-type integrals involving product s of Schur functions using a determinantal approach\, and point out severa l combinatorial connections\, most notably to hook length formulas. In the second part we will explain how further generalizations of the determinan ts lead to hypergeometric functions with Jack polynomial arguments on the one hand and number theoretic results on the other.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Greta Panova (University of Southern California) DTSTART:20201126T040000Z DTEND:20201126T050000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/28 DESCRIPTION:Title: The mysterious Kronecker coefficients\nby Greta Panova (Universi ty of Southern California) as part of IMSc algebraic combinatorics seminar \n\n\nAbstract\nEver since their definition in the 1930s\, as multipliciti es of irreducible symmetric group representations in the tensor product of two others\, the Kronecker coefficients have eluded our attempts to descr ibe them combinatorially or to compute them efficiently. Computational com plexity gives us the tools to formalize that mysteriousness. We will discu ss their "hardness" and show how to obtain some effective bounds despite t he lack of efficient formulas.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Volodymyr Mazorchuk (Uppsala University) DTSTART:20201203T093000Z DTEND:20201203T103000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/29 DESCRIPTION:Title: Bigrassmannian permutations and Verma modules\nby Volodymyr Mazo rchuk (Uppsala University) as part of IMSc algebraic combinatorics seminar \n\n\nAbstract\nIn this talk I will try to describe an unexpected connecti on\nbetween bigrassmannian permutations and the cokernel of inclusions\nbe tween Verma modules (over the special linear Lie algebra). An\napplication (and the original motivation) is a complete description of the first exte nsion space from a simple highest wegiht module to a Verma\nmodules.\n\nTh is is a report on a joint work with Hankyung Ko and Rafael Mrden.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Rekha Biswal (University of Edinburgh) DTSTART:20201217T083000Z DTEND:20201217T093000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/30 DESCRIPTION:Title: Macdonald polynomials and level two Demazure modules for affine $\\m athfrak{sl}_{n+1}$\nby Rekha Biswal (University of Edinburgh) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nAn important result due to Sanderson and Ion says that characters of level one Demazure modul es are specialized Macdonald polynomials. In this talk\, I will introduce a new class of symmetric polynomials indexed by a pair of dominant weights of $\\mathfrak{sl}_{n+1}$ which is expressed as linear combination of spe cialized symmetric Macdonald polynomials with coefficients defined recursi vely. These polynomials arose in my own work while investigating the chara cters of higher level Demazure modules. Using representation theory\, we w ill see that these new family of polynomials interpolate between character s of level one and level two Demazure modules for affine $\\mathfrak{sl}_{ n+1}$ and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayanthi Char i\, Peri Shereen and Jeffrey Wand.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Pantangi Venkata Raghu Tej (SUSTech\, Shenzen) DTSTART:20201224T093000Z DTEND:20201224T103000Z DTSTAMP:20260422T212604Z UID:imsc-algcomb/31 DESCRIPTION:Title: EKR problems for permutation groups\nby Pantangi Venkata Raghu T ej (SUSTech\, Shenzen) as part of IMSc algebraic combinatorics seminar\n\n \nAbstract\nErdős-Ko-Rado (EKR) theorem is a classical result in extremal set theory. It characterizes the largest collection of pair-wise $k$-subs ets of an $n$-set. An active line of research is to generalize this result to different objects. Let $G$ be a transitive permutation group on a set $\\Omega$. A subset $\\mathcal{F}\\subset G$ is said to be an intersecting set if any two permutations $g\,h\\in \\mathcal{F}$ agree on a point $\\o mega \\in \\Omega$\, that is\, $\\omega^{g}=\\omega^{h}$. Cosets of point stabilizers are natural examples of intersecting sets. An intersecting set is said to be a maximum intersecting set if it is of the maximum possible size. In view of the classical EKR theorem\, it is of interest to charact erize maximum intersecting sets. A group is said to satisfy the EKR proper ty if for every intersecting set $\\mathcal{F}$\, we have $|\\mathcal{F}|\ \leq|G_{\\omega}|$\, that is\, cosets of point stabilizers are maximum int ersecting sets. It is known that if $G$ is either Frobenius or $2$-transit ive\, it satisfies the EKR property. In this talk\, we will see that gener al transitive permutation groups are quite far from satisfying the EKR pro perty. In particular\, we show that even in the case of primitive groups\, there is no absolute constant $c$ such that $|\\mathcal{F}|\\leqslant c|G _\\omega|$. This is joint work with Cai Heng Li and Shu Jiao Song.\n LOCATION:https://researchseminars.org/talk/imsc-algcomb/31/ END:VEVENT END:VCALENDAR