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:20260404T095335Z UID:imsc-algcomb/1 DESCRIPTION:Title: Crystal for stable Grothendieck polynomials\, part 2\nby A nne Schilling (UC Davis) as part of IMSc algebraic combinatorics seminar\n \n\nAbstract\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-Hec ke monoid which we call ⋆-crystal. This crystal is a K-theoretic general ization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the ⋆ -crystal intertwines with the crystal on set-valued tableaux recently intr oduced by Monical\, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization in the 0-Hecke monoid to pairs of (transpos es of) semistandard Young tableaux and prove several properties about this new insertion\, in particular its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Arvind Ayyer (IISc\, Bangalore) DTSTART:20200422T093000Z DTEND:20200422T110000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/2 DESCRIPTION:Title: Combinatorics of an exclusion process driven by an asymmetric tracer\nby Arvind Ayyer (IISc\, Bangalore) as part of IMSc algebraic c ombinatorics seminar\n\n\nAbstract\nWe consider an exclusion process on a periodic one-dimensional lattice where all particles perform simple symmet ric exclusion except for a _tracer particle_\, which performs partially as ymmetric exclusion with forward and backward rates p and q respectively. T his process and its variants have been investigated starting with Ferrari\ , Goldstein and Lebowitz (1985) motivated by questions in statistical phys ics. We prove product formulas for stationary weights and exact formulas f or the nonequilibrium partition function in terms of combinatorics of set partitions. We will also compute the current\, and the density profile as seen by the test particle. Time permitting\, we will illustrate the ideas involved in performing asymptotic analysis. This talk is based on the pre print arXiv:2001.02425.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/2/ END:VEVENT BEGIN:VEVENT SUMMARY:G. Arunkumar (IISER Mohali) DTSTART:20200429T053000Z DTEND:20200429T070000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/3 DESCRIPTION:Title: Chromatic Symmetric Function of Graphs from Borcherds Lie Alge bra\nby G. Arunkumar (IISER Mohali) as part of IMSc algebraic combinat orics seminar\n\n\nAbstract\nLet $\\mathfrak g$ be a Borcherds algebra wit h the associated graph $G$. We prove tha\\\nt the chromatic symmetric func tion of $G$ can be recovered from the Weyl denominators of $\\mathfrak g$ and this gives a Lie theoretic proof of Stanley’s expression for chromat ic symmetric function in terms of power sum symmetric functions. Also\, th is gives an expression for the chromatic symmetric function of $G$ in term s of root multiplicities of $\\mathfrak g$. We prove a modified Weyl denom inator identity for Borcherds algebras which is an extension of the celebr ated classical Weyl denominator identity and this plays an important role in the proof our results. The absolute value of the linear coefficient of the chromatic polynomial of $G$ is known as the chromatic discriminant of $G$. As an application of our main theorem\, we prove that certain coeffic ients appearing in the above said expression of chromatic symmetric functi on is equal to the chromatic discriminant of $G$. Also\, we find a connect ion between the Weyl denominators and the $G$-elementary symmetric functio ns. Using this connection\, we give a Lie-theoretic proof of non-negativit y of coefficients of $G$-power sum symmetric functions.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Mike Zabrocki (York University) DTSTART:20200506T150000Z DTEND:20200506T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/4 DESCRIPTION:Title: Symmetric group characters as symmetric functions\nby Mike Zabrocki (York University) as part of IMSc algebraic combinatorics semina r\n\n\nAbstract\nI will present a basis of the symmetric functions whose e valuations are irreducible\ncharacters of the symmetric group in the same way that the evaluations of Schur\nfunctions are irreducible characters of the general linear group.  These symmetric\nfunctions are related to cha racter polynomials (that go back to a paper of\nFrobenius in 1904) but the y have the advantage that we are able to use the Hopf\nstructure of the sy mmetric functions to compute with them.  In addition\, they\nindicate tha t the combinatorics of Kronecker coefficients is governed by multiset\ntab leaux.  We use this basis to give a combinatorial interpretation for the tensor\nproducts of the form\n$$\\chi^{(n-|\\lambda|\,\\lambda)} \\otimes  \\chi^{(n-a_1\,a_1)} \n\\otimes \\chi^{(n-a_2\,a_2)} \\otimes \\cdots  \\otimes \\chi^{(n-a_r\,a_r)}$$\nwhere $\\lambda$ is a partition and $a _1\, a_2\, \\ldots\, a_r$ are non-negative\nintegers.\n\nThis is joint wor k with Rosa Orellana.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Nate Harman (The Univesity of Chicago) DTSTART:20200513T150000Z DTEND:20200513T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/5 DESCRIPTION:Title: Intermediate Algebraic Structure in the Restriction Problem\nby Nate Harman (The Univesity of Chicago) as part of IMSc algebraic com binatorics seminar\n\n\nAbstract\nThe restriction problem refers to unders tanding in a combinatorial sense the\ndecomposition of an irreducible repr esentation of GL_n as a representation of S_n. In\nthis talk\, I will dis cuss some of the intermediate algebraic structures that arise when\nstudyi ng this problem which constrain the symmetric group representations that a ppear and\n(hopefully) give some insight into the general problem. Things I will mention include: \nRepresentation stability\, the rook monoid\, th e group of monomial matrices\, and a certain\nsubalgebra of the universal enveloping algebra which seems to have interesting\ncombinatorial properti es.\n LOCATION:https://stable.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:20260404T095335Z UID:imsc-algcomb/6 DESCRIPTION:Title: A timed version of the plactic monoid\nby Amritanshu Prasa d (The Institute of Mathematical Sciences\, Chennai) as part of IMSc algeb raic combinatorics seminar\n\n\nAbstract\nLascoux and Schutzenberger intro duced the plactic monoid as a tool to prove the Littlewood-Richardson rule . The plactic monoid is the quotient of the free monoid on an ordered alph abet modulo Knuth relations. In this talk I will explain how their theory can be generalized to timed words\, which are words where each letter occu rs for a positive amount of time rather than discretely. This generalizati on gives an organic approach to piecewise-linear interpolations of corresp ondences 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://stable.researchseminars.org/talk/imsc-algcomb/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Deniz Kus (Ruhr-University Bochum) DTSTART:20200611T150000Z DTEND:20200611T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/7 DESCRIPTION:Title: Quiver varieties and their combinatorial crystal structure \nby Deniz Kus (Ruhr-University Bochum) as part of IMSc algebraic combinat orics seminar\n\n\nAbstract\nThe aim of this talk is to describe combinato rially the crystal operators on the geometric realization of crystal bases in terms 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 combi natorics is in terms of Auslander Reiten quivers.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Inna Entova (Ben Gurion University) DTSTART:20200625T150000Z DTEND:20200625T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/8 DESCRIPTION:Title: Deligne categories and stable Kronecker coefficients\nby I nna Entova (Ben Gurion University) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nIn this talk\, I will present an application of th e theory of Deligne categories to the study of Kronecker coefficients.\n\n

Kronecker coefficients are structural constants for the category $\\mathrm{Rep}(S_n)$ of finite-dimensional representations of the symmetri c g\\\nroup\; namely\, given three irreducible representations $\\mu\, \\t au\, \\lambda$ of $S_n$\, the Kronecker coefficient $\\mathrm{Kron}( \\lam bda\, \\mu\, \\tau)$ i\\\ns the multiplicity of $\\lambda$ inside $\\mu \ \otimes \\tau$.\n The study of Kronecker coefficients has been desc ribed as "one of the main problems in the combinatorial representation the ory of the symmetri\\\nc group"\, yet very little is known about them.

\n\n

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

\n\n

It turns out that the stable Kron ecker coefficients appear naturally as structural constants in the\n Deligne categories $\\mathrm{Rep}(S_t)$\, which are interpolations of the categories $\\mathrm{Rep}(S_n)$ to complex $t$. I\n will explain this phenomenon\, and show that the categorical properties of $\\mathrm{Rep}(S _t)$ allow us not only to\n recover known properties of the stable Kronecker coefficients\, but also obtain new identities.

\n

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

\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Jeanne Scott (Universidad de los Andes) DTSTART:20200604T150000Z DTEND:20200604T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/9 DESCRIPTION:Title: What's the right notion of content for the Young-Fibonacci lat tice?\nby Jeanne Scott (Universidad de los Andes) as part of IMSc alge braic combinatorics seminar\n\n\nAbstract\nThe Young-Fibonacci lattice $\\ Bbb{YF}$ is a ranked lattice invented by R. Stanley as an example of a dif ferential poset\; a nice consequence of this feature is that saturated cha ins (which 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 simp le Coxeter-like presentation. The representation theory of these algebras strongly parallels the story of the symmetric groups $S(n)$ --- in particu lar each element w of rank $\\mathrm{rk}(w) = n$ in the $\\Bbb{YF}$ lattic e corresponds to an irreducible representation $V(w)$ of $\\frak{F}(n)$ wh ose basis is indexed by saturated chains in the $\\Bbb{YF}$ lattice ending at $w$. Furthermore there is a theory of $\\Bbb{YF}$-Schur functions obey ing a Littlewood-Richardson rule whose structure coefficients coincide wit h the induction product multiplicities for representations of the Okada a lgebras.\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 th e (maximal) commutative subalgebra of $\\frak{F}(n)$ generated by the cent ers $Z\\frak{F}(1)\, Z\\frak{F}(2)\, \\dots\, Z\\frak{F}(n)$. 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) ea ch $J(n)$ resides in $\\mathrm{GT}(n)$
\n\n (2) $J(1)\, \\dots\, J(n)$ generate $\\mathrm{GT}(n)$
\n\n (3) the sum $J(1) + \\cdots + J(n)$ res ides in $Z\\frak{F}(n)$
\n\n (4) each $J(k)$ acts diagonally on the irr educible representation $V(w)$ and its eigenvalue\, with respect to a basi s vector indexed by a saturated chain $u(0) \\lhd \\cdots \\lhd u(n)$\, de pend only on the covering relation $u(k-1) \\lhd u(k)$ in $\\Bbb{YF}$.
\n\n\nThis local eigenvalue $c(u \\lhd v)$ is called the content of coveri ng relation $u \\lhd v$ with respect to the choice of Jucys-Murphy generat ors. Keep in mind that there are many different systems of elements $J(n)$ satisfying properties (1)\, (2)\, (3)\, and (4). However\, not any assign ment of covering weights $c(u \\lhd v)$ can be realized as contents for su ch a system. Indeed a necessary condition requires that two saturated chai ns coincide if and only if the corresponding sequences of covering weights are equal\; 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 gr oups $S(n)$ together with its branching poset\, the Young lattice $\\Bbb{Y }$\, as a guide to impose further constraints. For example\, one might try determine a system of Jucys-Murphy elements by forcing the attending syst em of contents to satisfy a specialization formula for the $\\Bbb{YF}$-Sch ur functions in analogy with the principal specialization of classical Sch ur functions. This is work in progress.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Shubham Sinha (UCSD) DTSTART:20200618T150000Z DTEND:20200618T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/10 DESCRIPTION:Title: Random $t$-cores and hook lengths in random partitions\nb y Shubham Sinha (UCSD) as part of IMSc algebraic combinatorics seminar\n\n \nAbstract\nFix $t \\geq 2$. We first give an asymptotic formula for certa in sums of the number of $t$-cores. We then use this result to compute the distribution 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 distribu tion after 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 ap ply this result to show that the probability that $t$ divides the hook len gth of a uniformly random cell in a uniformly random partition equals $1/t $ in the limit. Finally\, we extend this result to all modulo classes of $ t$ using abacus representations for cores and quotients. This talk is base d on the arxiv preprint arXi v:1911.03135.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Sheila Sundaram (Pierrepont School\, Westport\, CT\, USA) DTSTART:20200702T150000Z DTEND:20200702T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/11 DESCRIPTION:Title: Plethystic inversion and representations of the symmetric gro up\nby Sheila Sundaram (Pierrepont School\, Westport\, CT\, USA) as pa rt of IMSc algebraic combinatorics seminar\n\n\nAbstract\nIn this talk we will survey the many instances of plethystic inversion that occur in the r epresentation 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 free Lie algebra\, appears here. We will discuss the equivalence\n of Cadogan's formula to Thrall's decomposition of the regular representation \, and to many other phenomena in\n a wide variety of contexts. New decomp ositions of the regular representation will be presented. Some of this\n m aterial appears in the following papers:\n\n arXiv:1803.09368\n\n arXiv:2003.10700.\n\n arXiv:2006.01896\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/11/ END:VEVENT BEGIN:VEVENT SUMMARY:Digjoy Paul (IMSc\, Chennai) DTSTART:20200716T150000Z DTEND:20200716T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/12 DESCRIPTION:Title: Quasi p-Steinberg Character for Symmetric\, Alternating Group s and their Double Covers\nby Digjoy Paul (IMSc\, Chennai) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nGiven a finite group o f Lie type in characteristic $p$\,  Steinberg constructed a distinguishe d ordinary representation of dimension equals to the\nthe cardinality of a Sylow-$p$-subgroup and whose character\, which is now known as $p$-Ste inberg character\, vanishes except at $p$-regular elements. \n \nThe f ollowing question was raised by W. Feit\, and was answered by M. R. Daraf sheh for the alternating group or the projective special linear group:\n  "Let $G$ be a finite simple group of order divisible by the prime $p$\, an d suppose that $G$ has a $p$-Steinberg character. Does it follow that $G$ is a\nsemisimple group of Lie type in characteristic $p$?"\n\nThis motivat es us to define Quasi $p$-Steinberg character for finite groups.\nAn ir reducible 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 Chara cters of Symmetric as well as  Alternating groups and their double covers . On the\nway\, we also answer a question\, similar to Feit\, asked by  D ipendra Prasad.\nThis talk is based on ongoing work with Pooja Singla.\n\ nReferences:\n\n1. J. E. Humphreys\, The Steinberg representation\,1987.\ n\n2. W. Feit\, Extending Steinberg Characters\,1993.\n\n3. M. R. Darafs heh\, $p$-Steinberg Characters of Alternating and Projective Special Linea r Groups 1995.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Digjoy Paul (IMSc\, Chennai) DTSTART:20200806T150000Z DTEND:20200806T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/13 DESCRIPTION:Title: Quasi $p$-Steinberg Characters of double covers of Symmetric and Alternating groups\nby Digjoy Paul (IMSc\, Chennai) as part of IMS c algebraic combinatorics seminar\n\n\nAbstract\nAn irreducible character of a finite group $G$ is called Quasi $p$-Steinberg for a prime $p$ if it takes non-zero value on every $p$-regular element of $G$.\n\nIn this talk\ , we shall recall some combinatorial aspects of the representation theory of double covers of Symmetric and Alternating groups. Then we discuss the existence of Quasi\n$p$-Steinberg Characters of those groups. This talk i s based on ongoing work with Pooja Singla.\n\n\nSuggested readings:\n\n1. A. O. Morris\, The spin representation of the symmetric group\, Proc. Lond on Math. Soc. (3)\, 12 (1962).\n\n2. J. R. Stembridge\, Shifted tableaux a nd the projective representations of the symmetric groups. Adv. in Math. 7 4 (1989).\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Ryan Vinroot (William and Mary) DTSTART:20200813T150000Z DTEND:20200813T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/14 DESCRIPTION:Title: Generating Functions for Involutions and Character Degree Sum s in Finite 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 irred ucible representations of $G$ may be defined over the reals if and only if \nthe character degree sum of $G$ is equal to the number of involutions of $G$.  We use\nthis result and generatingfunctionology to study the real representations of finite\ngroups of Lie type\, and to obtain some new com binatorial identities.  We will begin with\nexamples of Weyl groups\, the n discuss joint work with Jason Fulman on finite general\nlinear and unita ry groups\, and then give more recent results for finite symplectic and\no rthogonal groups.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Anupam Kumar Singh (IISER Pune) DTSTART:20200820T043000Z DTEND:20200820T053000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/15 DESCRIPTION:Title: Asymptotics of powers in finite reductive groups\nby Anup am Kumar Singh (IISER Pune) as part of IMSc algebraic combinatorics semina r\n\n\nAbstract\nLet $G$ be a connected reductive group defined over a fin ite field $\\mathbf F_q$. Fix an integer $M >1$\, and consider the power m ap $x$ going to $x^M$ on G. We denote the image of $G(\\mathbf F_q)$ under this map by $G(\\mathbf F_q)^M$ and estimate what proportion of regular s emisimple\, semisimple and regular elements of $G(\\mathbf F_q)$ it contai ns. We prove that as q tends to infinity\, all of these proportions are eq ual and provide a formula for the same. We also calculate this more explic itly for the groups $GL(n\, q)$ and $U(n\, q)$.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Sami H Assaf (University of Southern California) DTSTART:20200903T033000Z DTEND:20200903T043000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/16 DESCRIPTION:Title: Pieri rules for polynomials\nby Sami H Assaf (University of Southern California) as part of IMSc algebraic combinatorics seminar\n\ n\nAbstract\nSchur functions are an amazing basis of symmetric functions o riginally defined as characters of irreducible modules for of $GL_n$. The Pieri rule for the product of a Schur function and a single row Schur func tion is a multiplicity-free branching rule with a beautiful combinatorial interpretation in terms of adding boxes to a Young diagram. Key polynomial s are an interesting basis of the polynomial ring originally defined as ch aracters of submodules for irreducible $GL_n$ modules under the action of upper triangular matrices. In joint work with Danjoseph Quijada\, we give a Pieri rule for the product of a key polynomial and a single row key poly nomial. While this formula has signs\, it is multiplicity-free and has an interpretation in terms of adding balls to a key diagram\, perhaps after d ropping some balls down. Time permitting\, I’ll give applications to Sch ubert polynomials where the signs cancel to give a positive Pieri formula. \n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Subhajit Ghosh (Indian Institute of Science) DTSTART:20200827T053000Z DTEND:20200827T063000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/17 DESCRIPTION:Title: Total variation cutoff for random walks on some finite groups \nby Subhajit Ghosh (Indian Institute of Science) as part of IMSc alge braic combinatorics seminar\n\n\nAbstract\nThis presentation will be on th e mixing times for three random walk models. Specifically these are the ra ndom walks on the alternating group\, the group of signed permutations and the complete monomial group. The details for the models are given below:< br>\n The random walk on the alternating group: We investigate the properties 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 transpose top-$2$ with random shuffle. We find the spectrum of the transition matrixof this shuffle. We obtain the sharp mixing time by prov ing 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 permutations: We consider a random walk on the hyperoctahedral group $B_n$ generated 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 ran dom shuffle on $B_n$. We find the spectrum of the transition probabili ty matrix for this shuffle. We prove that this shuffle exhibits the total variation cutoff phenomenon with cutoff time $n\\log n$. Furthermore\, we show that a similar random walk on the demihyperoctahedral group $D_n$ gen erated by the identity signed permutation and the signed permutations of t he form $(i\,n)$ and $(-i\,n)$ for $1\\leq i< n$ also has a cutoff at $\\l eft(n-\\frac{1}{2}\\right)\\log n$.
\nThe random walk on the complet e monomial group: Let $G_1\\subseteq\\cdots\\subseteq G_n \\subseteq\\ cdots $ be a sequence of finite groups with $|G_1|>2$. We study the proper ties of a random walk on the complete monomial group $G_n\\wr S_n$ generat ed by the elements of the form $(\\text{e}\,\\dots\,\\text{e}\,g\;\\text{i d})$ and $(\\text{e}\,\\dots\,\\text{e}\,g^{-1}\,\\text{e}\,\\dots\,\\text {e}\,g\;(i\,n))$ for $g\\in G_n\,\\\;1\\leq i< n$. We call this the war p-transpose top with random shuffle on $G_n\\wr S_n$. We find the spec trum of the transition probability matrix for this shuffle. We prove that the mixingtime 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://stable.researchseminars.org/talk/imsc-algcomb/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Krishnan Rajkumar (Jawaharlal Nehru University) DTSTART:20200910T083000Z DTEND:20200910T093000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/18 DESCRIPTION:Title: Generalizations of the Selberg integral and combinatorial con nections\nby Krishnan Rajkumar (Jawaharlal Nehru University) as part o f IMSc algebraic combinatorics seminar\n\n\nAbstract\nWe'll briefly recall the history of the Selberg Integral and several variants. We'll also go t hrough the proof of some of them like Aomoto's integral before focusing on known and possibly new integrals involving Schur polynomials and Jack pol ynomials. We shall note the implications that these integrals seem to coun t (after a suitable normalization) the number of standard young tableaux o f skew shapes\, before conjecturing the existence of several Naruse-type h ook length formulas. Finally we will explain how these integrals arise in number theoretic problems.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Sumana Hauti (IISc\, Bangalore) DTSTART:20200917T083000Z DTEND:20200917T100000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/19 DESCRIPTION:Title: On Schur multipliers and projective representations of Heisen berg groups\nby Sumana Hauti (IISc\, Bangalore) as part of IMSc algebr aic combinatorics seminar\n\n\nAbstract\nThe study of projective represent ations has a long history starting with the pioneering work of Schur for f inite groups which involves understanding homomorphisms from a group into the projective linear groups. In this study\, an important role is played by a group called the Schur multiplier. In this talk\, we shall describe t he Schur 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 fi nite-dimensional complex projective irreducible representations. This is j oint work with Pooja Singla.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Rijubrata Kundu (IISER Pune) DTSTART:20200924T150000Z DTEND:20200924T163000Z DTSTAMP:20260404T095335Z 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://stable.researchseminars.org/talk/imsc-algcomb/20/ END:VEVENT BEGIN:VEVENT SUMMARY:S Viswanath (The Institute of Mathematical Sciences\, Chennai) DTSTART:20201001T083000Z DTEND:20201001T093000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/21 DESCRIPTION:Title: Saturation for refined Littlewood-Richardson coefficients-I\nby S Viswanath (The Institute of Mathematical Sciences\, Chennai) as p art of IMSc algebraic combinatorics seminar\n\n\nAbstract\nThe Littlewood- Richardson (LR) coefficients are the multiplicities of irreducible represe ntations occurring in the tensor product of two irreducible polynomial rep resentations of GL_n. To each permutation 'w' in S_n\, we associate a 'w-r efinement' of the LR coefficients. These correspond to multiplicities in t he so-called Kostant-Kumar submodules of the tensor product\, or equivalen tly of multiplicities in "excellent filtrations" of Demazure modules. We p rove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231-avoiding\, by adapting the proof via hives of the classical satura tion conjecture due to Knutson-Tao. This is a report of work-in-progress w ith Mrigendra Singh Kushwaha and KN Raghavan. This talk will span two semi nar days (Oct 1 and 8). In the first part\, we describe the setting of the problem and the result. In the second part\, we recall the key steps in t he Knutson-Tao proof of the saturation conjecture via hives and indicate h ow it can be adapted to our case.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/21/ END:VEVENT BEGIN:VEVENT SUMMARY:S Viswanath (The Institute of Mathematical Sciences\, Chennai) DTSTART:20201015T053000Z DTEND:20201015T063000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/22 DESCRIPTION:Title: Saturation for refined Littlewood-Richardson coefficients-2\nby S Viswanath (The Institute of Mathematical Sciences\, Chennai) as p art of IMSc algebraic combinatorics seminar\n\n\nAbstract\nThe Littlewood- Richardson (LR) coefficients are the multiplicities of irreducible represe ntations occurring in the tensor product of two irreducible polynomial rep resentations of GL_n. To each permutation 'w' in S_n\, we associate a 'w-r efinement' of the LR coefficients. These correspond to multiplicities in t he so-called Kostant-Kumar submodules of the tensor product\, or equivalen tly of multiplicities in "excellent filtrations" of Demazure modules. We p rove a saturation theorem for these w-refinements when 'w' is 312-avoiding or 231-avoiding\, by adapting the proof via hives of the classical satura tion conjecture due to Knutson-Tao. This is a report of work-in-progress w ith Mrigendra Singh Kushwaha and KN Raghavan. This talk will span two semi nar days (Oct 1 and 8). In the first part\, we describe the setting of the problem and the result. In the second part\, we recall the key steps in t he Knutson-Tao proof of the saturation conjecture via hives and indicate h ow it can be adapted to our case.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/22/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science) DTSTART:20201112T083000Z DTEND:20201112T093000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/23 DESCRIPTION:Title: Polya frequency sequences: analysis meets algebra\nby Apo orva Khare (Indian Institute of Science) as part of IMSc algebraic combina torics seminar\n\n\nAbstract\nI will provide an introduction to the theory of Polya frequency (PF) sequences. The first half includes examples (incl uding log-concave/unimodal sequences\, Hilbert series) and classical resul ts on generating functions of PF sequences (with some proofs\, and one rel ated Hypothesis). In the second half\, I discuss connections from total po sitivity to old and new phenomena involving symmetric functions. (Partly j oint with Terence Tao.)\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/23/ END:VEVENT BEGIN:VEVENT SUMMARY:Ajmain Yamin (Stony Brook) DTSTART:20201022T150000Z DTEND:20201022T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/24 DESCRIPTION:Title: Filtering Grassmannian Cohomology via k-Schur Functions\n by Ajmain Yamin (Stony Brook) as part of IMSc algebraic combinatorics semi nar\n\n\nAbstract\nThis talk concerns the cohomology rings of complex Gras smannians. In 2003\, Reiner and Tudose conjectured the form of the Hilbert series for certain subalgebras of these cohomology rings. We build on the ir work in two ways. First\, we conjecture two natural bases for these sub algebras that would imply their conjecture using notions from the theory o f k-Schur functions. Second\, we formulate an analogous conjecture for Lag rangian Grassmannians.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/24/ END:VEVENT BEGIN:VEVENT SUMMARY:Rosa Orellana (Dartmouth) DTSTART:20201029T150000Z DTEND:20201029T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/25 DESCRIPTION:Title: The Howe duality for the symmetric group\nby Rosa Orellan a (Dartmouth) as part of IMSc algebraic combinatorics seminar\n\n\nAbstrac t\nClassical Howe dualities provide a representation theoretical framework for\nclassical invariant theory.  In the classical Howe duality\,\n$GL_n (\\mathbb{C})$ is dual to $GL_k(\\mathbb{C})$ when acting on the\npolynomi al 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 joint work with Mike Zabrocki.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/25/ END:VEVENT BEGIN:VEVENT SUMMARY:Amrutha P (IISER Thiruvananthapuram) DTSTART:20201105T150000Z DTEND:20201105T163000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/26 DESCRIPTION:Title: On the determinant of representations of generalized symmetri c groups $Z_r\\wr S_n$.\nby Amrutha P (IISER Thiruvananthapuram) as pa rt of IMSc algebraic combinatorics seminar\n\n\nAbstract\nThe problem of e numeration of the number of irreducible representations of the symmetric g roup with a non trivial determinant was first considered by L. Solomon and later posed by Stanley in his book. Recently\, several authors have chara cterized and counted the number of irreducible representations of a given finite group with nontrivial determinant. Motivated by these results\, we are interested in the study of the determinant of irreducible representati ons of the generalized symmetric groups\, $Z_r \\wr S_n$. We give an expli cit formula to compute the determinant of an irreducible representation of $Z_r \\wr S_n$. Also\, for a given integer $n$\, and a prime number $r$ a nd $\\zeta$ 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 nu mber of irreducible representations of $Z_r \\wr S_n$ whose determinant is $\\zeta$.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/26/ END:VEVENT BEGIN:VEVENT SUMMARY:Krishnan Rajkumar (JNU) DTSTART:20201119T083000Z DTEND:20201119T093000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/27 DESCRIPTION:Title: Selberg integrals involving Schur polynomials and combinatori al connections\nby Krishnan Rajkumar (JNU) as part of IMSc algebraic c ombinatorics seminar\n\n\nAbstract\nIn the second take on this topic\, we will skip the history and focus on the combinatorial connections. In the f irst part of the talk we will (re)derive Selberg-type integrals involving products of Schur functions using a determinantal approach\, and point out several combinatorial connections\, most notably to hook length formulas. In the second part we will explain how further generalizations of the det erminants lead to hypergeometric functions with Jack polynomial arguments on the one hand and number theoretic results on the other.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/27/ END:VEVENT BEGIN:VEVENT SUMMARY:Greta Panova (University of Southern California) DTSTART:20201126T040000Z DTEND:20201126T050000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/28 DESCRIPTION:Title: The mysterious Kronecker coefficients\nby Greta Panova (U niversity of Southern California) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nEver since their definition in the 1930s\, as multi plicities of irreducible symmetric group representations in the tensor pro duct of two others\, the Kronecker coefficients have eluded our attempts t o describe them combinatorially or to compute them efficiently. Computatio nal complexity gives us the tools to formalize that mysteriousness. We wil l discuss their "hardness" and show how to obtain some effective bounds de spite the lack of efficient formulas.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/28/ END:VEVENT BEGIN:VEVENT SUMMARY:Volodymyr Mazorchuk (Uppsala University) DTSTART:20201203T093000Z DTEND:20201203T103000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/29 DESCRIPTION:Title: Bigrassmannian permutations and Verma modules\nby Volodym yr Mazorchuk (Uppsala University) as part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nIn this talk I will try to describe an unexpected c onnection\nbetween bigrassmannian permutations and the cokernel of inclusi ons\nbetween Verma modules (over the special linear Lie algebra). An\nappl ication (and the original motivation) is a complete description of the fir st extension space from a simple highest wegiht module to a Verma\nmodules .\n\nThis is a report on a joint work with Hankyung Ko and Rafael Mrden.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/29/ END:VEVENT BEGIN:VEVENT SUMMARY:Rekha Biswal (University of Edinburgh) DTSTART:20201217T083000Z DTEND:20201217T093000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/30 DESCRIPTION:Title: Macdonald polynomials and level two Demazure modules for affi ne $\\mathfrak{sl}_{n+1}$\nby Rekha Biswal (University of Edinburgh) a s part of IMSc algebraic combinatorics seminar\n\n\nAbstract\nAn important result due to Sanderson and Ion says that characters of level one Demazur e modules are specialized Macdonald polynomials. In this talk\, I will int roduce 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 specialized symmetric Macdonald polynomials with coefficients defined recursively. These polynomials arose in my own work while investigating th e characters of higher level Demazure modules. Using representation theory \, we will see that these new family of polynomials interpolate between ch aracters of level one and level two Demazure modules for affine $\\mathfra k{sl}_{n+1}$ and give rise to new results in the representation theory of current algebras as a corollary. This is based on joint work with Vyjayant hi Chari\, Peri Shereen and Jeffrey Wand.\n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/30/ END:VEVENT BEGIN:VEVENT SUMMARY:Pantangi Venkata Raghu Tej (SUSTech\, Shenzen) DTSTART:20201224T093000Z DTEND:20201224T103000Z DTSTAMP:20260404T095335Z UID:imsc-algcomb/31 DESCRIPTION:Title: EKR problems for permutation groups\nby Pantangi Venkata Raghu Tej (SUSTech\, Shenzen) as part of IMSc algebraic combinatorics semi nar\n\n\nAbstract\nErdős-Ko-Rado (EKR) theorem is a classical result in e xtremal set theory. It characterizes the largest collection of pair-wise $ k$-subsets 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 inter secting set if any two permutations $g\,h\\in \\mathcal{F}$ agree on a poi nt $\\omega \\in \\Omega$\, that is\, $\\omega^{g}=\\omega^{h}$. Cosets of point stabilizers are natural examples of intersecting sets. An intersect ing set is said to be a maximum intersecting set if it is of the maximum p ossible size. In view of the classical EKR theorem\, it is of interest to characterize maximum intersecting sets. A group is said to satisfy the EKR property if for every intersecting set $\\mathcal{F}$\, we have $|\\mathc al{F}|\\leq|G_{\\omega}|$\, that is\, cosets of point stabilizers are maxi mum intersecting sets. It is known that if $G$ is either Frobenius or $2$- transitive\, it satisfies the EKR property. In this talk\, we will see tha t general transitive permutation groups are quite far from satisfying the EKR property. In particular\, we show that even in the case of primitive g roups\, there is no absolute constant $c$ such that $|\\mathcal{F}|\\leqsl ant c|G_\\omega|$. This is joint work with Cai Heng Li and Shu Jiao Song.\ n LOCATION:https://stable.researchseminars.org/talk/imsc-algcomb/31/ END:VEVENT END:VCALENDAR