BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Pritam Majumder (Tata Institute of Fundamental Research\, Mumbai) DTSTART:20210219T093000Z DTEND:20210219T103000Z DTSTAMP:20260422T212604Z UID:ARCSIN/1 DESCRIPTION:Title: O n characterizing line graphs of hypergraphs\nby Pritam Majumder (Tata Institute of Fundamental Research\, Mumbai) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric functions in INdia\n\n\nAbst ract\nA hypergraph is given by a finite set of vertices together with a co llection of its subsets\, called edges\, of that set. A hypergraph is call ed k-uniform if all its edges have the same size k. The line graph of a k- uniform hypergraph is its edge-to-vertex dual graph\, namely\, its vertice s bijectively correspond to the edges of the hypergraph and there is an ed ge between two vertices of the line graph if the corresponding edges in th e hypergraph have non zero intersection. The characterization of line grap hs of 2-uniform hypergraphs (graphs) have been extensively studied. The ch aracterization of line graphs of k-uniform hypergraphs for k>2 is poorly u nderstood. Partial results exist for linear hypergraphs (intersection of a ny two edges is at most 1 vertex). We study the problem of \ncharacterizin g line graphs of k-uniform hypergraphs with bounded pair-degree by a finit e class of forbidden subgraphs. We show that such a characterization is po ssible if we consider line graphs with certain minimum edge-degree. Time p ermitting\, we shall discuss about some other reconstruction problems for hypergraphs\, namely\, characterizing degree sequences of hypergraphs and characterizing face numbers (f-vectors) of simplicial complexes.\n LOCATION:https://researchseminars.org/talk/ARCSIN/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Sridhar P Naryanan (The Institute of Mathematical Sciences\, Chenn ai) DTSTART:20210319T093000Z DTEND:20210319T103000Z DTSTAMP:20260422T212604Z UID:ARCSIN/2 DESCRIPTION:Title: M ultiplicity of trivial and sign representations of $S_n$ in hook-shaped re presentations of $GL_n$.\nby Sridhar P Naryanan (The Institute of Math ematical Sciences\, Chennai) as part of ARCSIN - Algebra\, Representations \, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nLet $W_\\ lambda$ be an irreducible representation of $GL_n$ (for\npartition $\\lamb da$ with $\\leq n$ parts). Let $V_\\mu$ be an irreducible\nrepresentation of $S_n$ (for partition $\\mu \\vdash n$). Then $$W_\\lambda=\n\\sum_{\\mu \\vdash n} r_{\\lambda \\mu} V_\\mu.$$\nThe coefficients $r_{\\lambda\\mu }$ are the restriction coefficients. The\nrestriction problem is to find c ombinatorial objects that these coefficients\ncount. We find such objects when $\\lambda$ is the hook shape and $\\mu=(n)$\nor $\\mu= (1^n)$ using t he theory of character polynomials and a simple\nsign-reversing involution .\n LOCATION:https://researchseminars.org/talk/ARCSIN/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Souvik Pal (Harish Chandra Research Institute\, Allahabad) DTSTART:20210326T093000Z DTEND:20210326T103000Z DTSTAMP:20260422T212604Z UID:ARCSIN/3 DESCRIPTION:Title: L evel zero integrable modules with finite-dimensional weight spaces for the graded Lie tori\nby Souvik Pal (Harish Chandra Research Institute\, Allahabad) as part of ARCSIN - Algebra\, Representations\, Combinatorics a nd Symmetric functions in INdia\n\n\nAbstract\nAn important problem in the representation theory of affine and toroidal Lie algebras is to classify all possible irreducible integrable modules with finite-dimensional weight spaces. The centres of both affine and toroidal Lie algebras are spanned by finitely many elements. If all these central elements act trivially on a module\, we say that the representation has level zero\, otherwise it is said to have non-zero level. The classification of these irreducible inte grable modules with finite-dimensional weight spaces over the affine Kac-M oody algebras (both twisted and untwisted) have been completely settled by V. Chari and A. Pressley. This was subsequently generalized by S. Eswara Rao for the (untwisted) toroidal Lie algebras. Recently\, the aforemention ed irreducible integrable modules of non-zero level have been classified f or a more general class of Lie algebras\, namely the graded Lie tori\, whi ch are multivariable generalizations of twisted affine Kac-Moody algebras. In this talk\, I shall address the mutually exclusive problem and hencefo rth classify all the level zero irreducible integrable modules with finite -dimensional weight spaces for this graded Lie tori.\n LOCATION:https://researchseminars.org/talk/ARCSIN/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Hassain Maliyekkal (The Institute of Mathematical Sciences\, Chenn ai\, India) DTSTART:20210430T093000Z DTEND:20210430T103000Z DTSTAMP:20260422T212604Z UID:ARCSIN/4 DESCRIPTION:Title: R epresentations of Compact Special Linear Groups of Degree Two\nby Hass ain Maliyekkal (The Institute of Mathematical Sciences\, Chennai\, India) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetri c functions in INdia\n\n\nAbstract\nLet $\\mathcal{O}$ be the ring of inte gers of a non-Archimedean local field and $\\wp$ be its maximal ideal. In this talk\, our focus is on the construction of the continuous complex irr educible representations of the group $\\mathrm{SL}_2(\\mathcal{O})$ and t o describe their representation growth. We will also discuss some results about group algebras of $\\mathrm{SL}_2(\\mathcal{O}/{\\wp}^r)$ for large $r$.\n LOCATION:https://researchseminars.org/talk/ARCSIN/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Digjoy Paul (Tata Institute of Fundamental Research) DTSTART:20211028T060000Z DTEND:20211028T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/5 DESCRIPTION:Title: S ymmetric $q\,t$ Catalan polynomials\nby Digjoy Paul (Tata Institute of Fundamental Research) as part of ARCSIN - Algebra\, Representations\, Com binatorics and Symmetric functions in INdia\n\n\nAbstract\nThe $q\, t$-Cat alan functions $C_n(q\,t)$\, an $q\, t$- analogue of Catalan numbers\, wer e first introduced in connection with Macdonald polynomials and Garsia–H aiman’s theory of diagonal harmonics [1996] as certain rational function s in $q$ and $t$. Haglund [2003] and shortly after that\, Haiman announced two combinatorial interpretations of $C_n(q\,t)$ as a weighted sum over a ll Dyck paths. An open problem related to these polynomials is a combinato rial proof of its symmetry in $q$ and $t$.\n\nWe define two symmetric $q\, t$ Catalan polynomials on Dyck paths and provide proof of the symmetry by establishing an involution on plane trees. We also give a combinatorial pr oof of a result by Garsia et al. regarding parking functions and the numbe r of connected graphs. This is joint work with Joseph Pappe and Anne Schil ling.\n LOCATION:https://researchseminars.org/talk/ARCSIN/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Manjunath Krishnapur (Indian Institute of Science\, Bangalore) DTSTART:20211111T060000Z DTEND:20211111T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/6 DESCRIPTION:Title: F ree probability\, free convolution and associated combinatorics of non-cro ssing partitions\nby Manjunath Krishnapur (Indian Institute of Science \, Bangalore) as part of ARCSIN - Algebra\, Representations\, Combinatoric s and Symmetric functions in INdia\n\n\nAbstract\nThis is an expository ta lk on free probability\, which is a part of operator theory with very stro ng parallels to probability theory. In particular\, we shall focus on free convolution\, which is a binary operation on measures on the real line th at is different from the usual convolution that arises when one adds indep endent random variables. Getting rid of analysis and expressing everything algebraically\, the difference between the two forms of convolution arise s from the difference between the lattice of all set partitions of a finit e set and the lattice of all non-crossing set partitions of the same. We w ould also like to explain how free convolution arises in random matrix the ory (what are the eigenvalues of the sum of two large matrices?) and in as ymptotic representation theory of symmetric groups (what are the Littlewoo d-Richardson coefficients of large partitions?). \n\nNot much knowledge o f probability will be assumed. We shall refer to Bernoulli and Gaussian ra ndom variables\, independence and convolution\, central limit theorem\, ma inly to explain the analogies. The talk should be accessible to advanced u ndergraduates.\n LOCATION:https://researchseminars.org/talk/ARCSIN/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Bishal Deb (University College London) DTSTART:20211118T060000Z DTEND:20211118T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/7 DESCRIPTION:Title: A nalysing a strategy for a card guessing game via continuously increasing s ubsequences in multiset permutations\nby Bishal Deb (University Colleg e London) as part of ARCSIN - Algebra\, Representations\, Combinatorics an d Symmetric functions in INdia\n\n\nAbstract\nConsider the following card guessing game introduced by Diaconis and Graham (1981): there is a shuffle d deck of $mn$ cards with $n$ distinct cards numbered $1$ to $n$\, each ap pearing with multiplicity $m$. In each round\, the player has to guess the top card of the deck\, and is then told whether the guess was correct or not\, the top card is then discarded and then the game continues with the next card. This is known as the partial feedback model. The aim is to maxi mise the number of correct guesses. One possible strategy is the shifting strategy in which the player keeps guessing $1$ every round until the gues s is correct in some round\, and then keeps guessing $2$\, and then $3$ an d so on. We are interested in finding the expected score using this strate gy.\n\nWe can restate this problem as finding the expectation of the large st value of $i$ such that $123\\ldots i$ is a subsequence in a word formed using letters 1 to n where each letter occurs with multiplicity $m$. In t his talk\, we show that this number is $m+1 - 1/(m+2)$ plus an exponential error term. This confirms a conjecture of Diaconis\, Graham\, He and Spir o.\n\nThis talk will be at an interface between combinatorics\, probabilit y and analysis and will feature an unexpected appearance of the Taylor pol ynomials of the exponential function. This is based on joint work with Ale xander Clifton\, Yifeng Huang\, Sam Spiro and Semin Yoo.\n LOCATION:https://researchseminars.org/talk/ARCSIN/7/ END:VEVENT BEGIN:VEVENT SUMMARY:Nishu Kumari (Indian Institute of Science) DTSTART:20211125T060000Z DTEND:20211125T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/8 DESCRIPTION:Title: F actorization of Classical Characters twisted by Roots of Unity\nby Nis hu Kumari (Indian Institute of Science) as part of ARCSIN - Algebra\, Repr esentations\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract \nIn representation theory\, Schur polynomials are the characters of the i rreducible polynomial representations of the classical groups of type A\, namely $GL_n(\\mathbb{C})$. \nMotivated by a celebrated result of Kostant\ , D. Prasad considered factorization of Schur polynomials \nin $tn$ variab les\, for $t \\geq 2$\, a fixed positive integer\, \nspecialized to $(\\ex p(2 \\pi \\iota k/t) x_j)_{0 \\leq k \\leq t-1\, 1 \\leq j \\leq n}$ (Isra el J. Math.\, 2016). He characterized partitions for which these Schur pol ynomials are nonzero and showed that if the Schur polynomial is nonzero\, it factorizes into characters of smaller classical groups of Type A.\n\nWe generalize Prasad's result to the irreducible characters of classical gro ups \nof type B\, C and D\, namely $O_{2tn+1}(\\mathbb{C})\, \n\\Sp_{2tn}( \\mathbb{C})$ and $O_{2tn}(\\mathbb{C})$\, with the same specialization. \ nWe give a uniform approach for all cases. \nThe proof uses Cauchy-type de terminant formulas for these characters and involves a careful study of th e beta sets of partitions. This is joint work with Arvind Ayyer and is ava ilable at https://arxiv.org/abs/2109.11310.\n LOCATION:https://researchseminars.org/talk/ARCSIN/8/ END:VEVENT BEGIN:VEVENT SUMMARY:Hemanshu Kaul (Illinois Institute of Technology) DTSTART:20220106T050000Z DTEND:20220106T060000Z DTSTAMP:20260422T212604Z UID:ARCSIN/9 DESCRIPTION:Title: C hromatic polynomial and counting DP-colorings of graphs : Problems and pro gress\nby Hemanshu Kaul (Illinois Institute of Technology) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nIn 1912\, Birkhoff\, introduced the chromatic po lynomial of a graph $G$ that counts the number of proper colorings of $G$. List coloring\, introduced in the 1970s by Erdos among others\, is a natu ral generalization of ordinary coloring where each vertex has a restricted list of colors available to use on it. The list color function of a graph is a list coloring analogue of the chromatic polynomial that has been stu died since 1990.\n\nDP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent ye ars after its introduction by Dvorak and Postle in 2015. Intuitively\, DP- coloring is a variation on list coloring where each vertex in the graph st ill gets a list of colors\, but identification of which colors are differe nt can change from edge to edge. It is equivalent to the question of findi ng independent transversals in a (DP-)cover of a graph. In this talk\, we will introduce a DP-coloring analogue of the chromatic polynomial called t he DP color function\, ask several fundamental open questions about it\, a nd give an overview of the progress made on them. We show that while the D P color function behaves similar to the list color function and chromatic polynomial for some graphs\, there are also some surprising fundamental di fferences. \n\nThe results are based on joint work with Jeffrey Mudrock (C LC)\, as well as several groups of students.\n LOCATION:https://researchseminars.org/talk/ARCSIN/9/ END:VEVENT BEGIN:VEVENT SUMMARY:Alexander Yong (UIUC) DTSTART:20220113T123000Z DTEND:20220113T133000Z DTSTAMP:20260422T212604Z UID:ARCSIN/10 DESCRIPTION:Title: Newell-Littlewood numbers\nby Alexander Yong (UIUC) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric functions in INd ia\n\n\nAbstract\nThe Newell-Littlewood numbers are defined in terms of th e \nLittlewood-Richardson coefficients from algebraic combinatorics. Both \nappear in representation theory as tensor product multiplicities f or a\nclassical Lie group. This talk concerns the question: \n\n Which multiplicities are nonzero? \n\nIn 1998\, Klyachko established c ommon linear inequalities defining \nboth the eigencone for sums of Hermit ian matrices and the saturated \nLittlewood-Richardson cone. We prove some analogues of Klyachko's nonvanishing\nresults for the Newell-Littlewood n umbers.\n\nThis is joint work with Shiliang Gao (UIUC)\, Gidon Orelowitz ( UIUC)\, and\nNicolas Ressayre (Universite Claude Bernard Lyon I). The pres entation is based on\narXiv:2005.09012\, arXiv:2009.09904\, and arXiv:2107 .03152.\n LOCATION:https://researchseminars.org/talk/ARCSIN/10/ END:VEVENT BEGIN:VEVENT SUMMARY:Sunil Chebolu (Illinois State University) DTSTART:20220127T060000Z DTEND:20220127T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/12 DESCRIPTION:Title: How many units can a commutative ring have?\nby Sunil Chebolu (Illinoi s State University) as part of ARCSIN - Algebra\, Representations\, Combin atorics and Symmetric functions in INdia\n\n\nAbstract\nLaszlo Fuchs posed the following problem in 1960\, which remains open: classify the abelian groups occurring as the group of all units in a commutative ring. In this talk\, I will provide an elementary solution to a simpler\, related proble m: find all cardinal numbers occurring as the cardinality of the group of all units in a commutative ring. As a by-product\, we obtain a solution to Fuchs' problem for the class of finite abelian p-groups when p is an odd prime.\n LOCATION:https://researchseminars.org/talk/ARCSIN/12/ END:VEVENT BEGIN:VEVENT SUMMARY:Samrith Ram (IIIT Delhi) DTSTART:20220203T083000Z DTEND:20220203T093000Z DTSTAMP:20260422T212604Z UID:ARCSIN/13 DESCRIPTION:Title: Set Partitions\, Tableaux\, and Subspace Profiles under Regular Split Semi simple Matrices\nby Samrith Ram (IIIT Delhi) as part of ARCSIN - Algeb ra\, Representations\, Combinatorics and Symmetric functions in INdia\n\n\ nAbstract\nIn this talk we will introduce a family of polynomials $b_\\lam bda(q)$ indexed by integer partitions $\\lambda$. These polynomials arise from an intriguing connection between two classical combinatorial classes\ , namely set partitions and standard tableaux. The polynomials $b_\\lambda (q)$ can be derived from a new statistic on set partitions called the inte rlacing number which is a variant of the well-known crossing number of a s et partition. These polynomials also have several interesting specializati ons: $b_\\lambda(1)$ enumerates the number of set partitions of shape $\\ lambda$ and $b_\\lambda(0)$ counts the number of standard tableaux of shap e $\\lambda$ while $b_\\lambda(-1)$ equals the number of standard shifted tableaux of shape $\\lambda$ respectively. When $q$ is a prime power $b_\\ lambda(q)$ counts (up to factors of $q$ and $q-1$) the number of subspaces in a finite vector space that transform under a regular diagonal matrix i n a specified manner.\n\nThis is joint work with Amritanshu Prasad.\n LOCATION:https://researchseminars.org/talk/ARCSIN/13/ END:VEVENT BEGIN:VEVENT SUMMARY:Santosh Nadimpalli (IIT Kanpur) DTSTART:20220217T060000Z DTEND:20220217T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/14 DESCRIPTION:Title: On uniqueness of branching to fixed point Lie subalgebras\nby Santosh Nadimpalli (IIT Kanpur) as part of ARCSIN - Algebra\, Representations\, Co mbinatorics and Symmetric functions in INdia\n\n\nAbstract\nLet $\\mathfra k{g}$ be a complex semisimple Lie algebra and let\n $\\theta$ be a finite order automorphism of $\\mathfrak{g}$. We assume\n that any ${\\rm A}_{2 n}$-type $\\theta$-stable indecomposable ideal of\n $\\mathfrak{g}$ is si mple and any ${\\rm D}_k$\, ${\\rm A}_{2k+1}$ and\n ${\\rm E}_6$-type $\\ theta$-stable indecomposable ideal of\n $\\mathfrak{g}$ has length at mos t $2$. Let $\\mathfrak{g}_0$ be the\n fixed point subalgebra of $\\mathfr ak{g}$. In this talk\, for any\n irreducible finite dimensional represen tations $V_1$ and $V_2$ of\n $\\mathfrak{g}$\, we show that\n${\\rm res}_ {\\mathfrak{g}_0}V_1\\simeq \n{\\rm res}_{\\mathfrak{g}_0}V_2$ if and only if $V_2$ is isomorphic to\n$V_1^\\sigma$\, for some outer automorphism $\ \sigma$ of $\\mathfrak{g}$.\n LOCATION:https://researchseminars.org/talk/ARCSIN/14/ END:VEVENT BEGIN:VEVENT SUMMARY:Shraddha Srivastava (Uppsala University) DTSTART:20220505T060000Z DTEND:20220505T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/15 DESCRIPTION:Title: Diagram categories and reduced Kronecker coefficients\nby Shraddha Sri vastava (Uppsala University) as part of ARCSIN - Algebra\, Representations \, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nPartition algebras are a class of diagram algebras which naturally fit into a tower and the so called partition category provides a unified framework for the study of the algebras in the tower. The path algebra of the partition cat egory admits a triangular decomposition similar to a triangular decomposit ion of the universal enveloping algebra of a finite dimensional complex se misimple Lie algebra. In such a decomposition\, the direct sum of symmetri c group algebras plays a role analogous to Cartan subalgebra and this prov ides a natural approach to the representation theory of the partition cate gory. The tensor structure on the partition category induces a ring struct ure on the associated Grothendieck group. Reduced Kronecker coefficients f or symmetric groups appear as structure constants in the Grothendieck ring . \n\nIn this talk\, we discuss the partition category and its connection to reduced Kronecker coefficients (these are results of several authors). We introduce the multiparameter colored partition category where the Carta n subalgebra in the corresponding triangular decomposition is given by com plex reflection groups of type $G(r\,1\,n)$. The multiparameter colored pa rtition category also admits a tensor structure. If time permits\, we also relate the associated Grothendieck ring for this category with the ring o f symmetric functions. This talk is based on joint work with Volodymyr Maz orchuk.\n LOCATION:https://researchseminars.org/talk/ARCSIN/15/ END:VEVENT BEGIN:VEVENT SUMMARY:Ashish Mishra (Universidade Federal do Pará\, Belém) DTSTART:20220303T100000Z DTEND:20220303T110000Z DTSTAMP:20260422T212604Z UID:ARCSIN/16 DESCRIPTION:Title: The Jucys--Murphy elements\nby Ashish Mishra (Universidade Federal do Pará\, Belém) as part of ARCSIN - Algebra\, Representations\, Combinator ics and Symmetric functions in INdia\n\n\nAbstract\nThe representation the ory of a multiplicity free tower of finite-dimensional semisimple associat ive algebras is determined by the actions of Jucys--Murphy elements. These elements were discovered independently by Jucys and Murphy for the symme tric groups\, and later on\, these elements played an important role in t he development of spectral approach to the representation theory of symmet ric groups given by Okounkov and Vershik. The motivation for the spectral approach comes from the work of Gelfand and Tsetlin on the irreducible fin ite-dimensional modules of general linear Lie algebras. \n\nAfter a brief description of the history and fundamental properties of Jucys--Murphy ele ments\, our main objective in this seminar is to describe these elements a nd to study their applications in the representation theory of following a lgebras: (i) partition algebras for complex reflection groups\, (ii) rook monoid algebras\, and (iii) totally propagating partition algebras. The re sults presented in this seminar are joint work with Dr. Shraddha Srivastav a.\n LOCATION:https://researchseminars.org/talk/ARCSIN/16/ END:VEVENT BEGIN:VEVENT SUMMARY:Hugh r Thomas (Université du Québec à Montréal) DTSTART:20220428T123000Z DTEND:20220428T133000Z DTSTAMP:20260422T212604Z UID:ARCSIN/17 DESCRIPTION:Title: The Robinson--Schensted--Knuth correspondence via quiver representations\nby Hugh r Thomas (Université du Québec à Montréal) as part of ARCS IN - Algebra\, Representations\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nThe RSK correspondence is a multi-faceted jewel at th e heart of algebraic combinatorics. In one of its incarnations\, it is a p iecewise-linear bijection between an orthant and a much more complicated c one which controls the structure of a pair of semistandard tableaux of the same shape. I will explain a classic enumerative result of Stanley which suggests the existence of such a map\, and then explain a way to construct it which arises naturally out of the theory of representations of quivers . No knowledge of quiver representations will be assumed. This talk is bas ed on joint work with Al Garver and Becky Patrias.\n LOCATION:https://researchseminars.org/talk/ARCSIN/17/ END:VEVENT BEGIN:VEVENT SUMMARY:Arvind Ayyer (Indian Institute of Science\, Bangalore) DTSTART:20220311T083000Z DTEND:20220311T093000Z DTSTAMP:20260422T212604Z UID:ARCSIN/18 DESCRIPTION:Title: A multispecies totally asymmetric zero range process and Macdonald polynom ials\nby Arvind Ayyer (Indian Institute of Science\, Bangalore) as par t of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric func tions in INdia\n\nLecture held in Alladi Ramakrishnan Hall\, IMSc\, Chenna i.\n\nAbstract\nMacdonald polynomials are a remarkable family of symmetric functions that\nare known to have connections to combinatorics\, algebrai c geometry and\nrepresentation theory. Due to work of Corteel\, Mandelshta m and Williams\, it\nis known that they are related to the asymmetric simp le exclusion process\n(ASEP) on a ring.\n\nThe modified Macdonald polynomi als are obtained from the Macdonald\npolynomials using an operation called plethysm. It is natural to ask whether\nthe modified Macdonald polynomial s are related to some other particle\nsystem. In this talk\, we answer thi s question in the affirmative via a\nmultispecies totally asymmetric zero- range process (TAZRP). We also present\na Markov process on tableaux that projects to the TAZRP and derive formulas\nfor stationary probabilities an d certain correlations. This is joint work\nwith Olya Mandelshtam and Jame s Martin.\n\nThis will be a hybrid event. Note that the usual Zoom link fo r the seminar is not valid.\n LOCATION:https://researchseminars.org/talk/ARCSIN/18/ END:VEVENT BEGIN:VEVENT SUMMARY:Nicolas Libedinsky (Universidad de Chile) DTSTART:20220526T110000Z DTEND:20220526T120000Z DTSTAMP:20260422T212604Z UID:ARCSIN/19 DESCRIPTION:Title: Introsurvey of representation theory\nby Nicolas Libedinsky (Universid ad de Chile) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nWe will give a summary of the paper "Introsurvey of representation theory". We will start with clas sical Schur-Weyl duality\, then introduce Iwahori-Hecke algebra and Soerge l bimodules. We will finish with some of the fascinating theorems and conj ectures around these objects.\n LOCATION:https://researchseminars.org/talk/ARCSIN/19/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science) DTSTART:20220520T060000Z DTEND:20220520T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/20 DESCRIPTION:Title: Higher order Verma modules\, and a positive formula for all highest weight modules - talk 1\nby Apoorva Khare (Indian Institute of Science) as p art of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric fu nctions in INdia\n\n\nAbstract\nWe study weights of highest weight modules $V$ over a Kac-Moody algebra $\\mathfrak{g}$ (one may assume this to be $ \\mathfrak{sl}_n$ throughout the talk\, without sacrificing novelty). We b egin with several positive weight-formulas for arbitrary non-integrable si mple modules\, and mention the equivalence of several "first order" data t hat helps prove these formulas. We then discuss the notion of "higher orde r holes" in the weights\, and use these to present two positive weight-for mulas for arbitrary modules $V$. One of these is in terms of "higher order Verma modules". (Joint with G.V.K. Teja and with Gurbir Dhillon.)\n LOCATION:https://researchseminars.org/talk/ARCSIN/20/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science) DTSTART:20220602T060000Z DTEND:20220602T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/21 DESCRIPTION:Title: Higher order Verma modules\, and a positive formula for all highest weight modules - talk 2\nby Apoorva Khare (Indian Institute of Science) as p art of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric fu nctions in INdia\n\n\nAbstract\nWe study weights of highest weight modules $V$ over a Kac-Moody algebra $\\mathfrak{g}$ (one may assume this to be $ \\mathfrak{sl}_n$ throughout the talk\, without sacrificing novelty). We b egin by recalling the notation\, "higher order holes"\, and "higher order Verma modules" (along with their universal property\, via examples). We th en recall our result from last time: the weights of any highest weight mod ule equal the weights of its "higher order" Verma cover.\n\nNext\, we defi ne the higher order category $\\mathcal{O}^\\mathcal{H}$\, and recall some properties in the zeroth and first order cases (work of Bernstein–Gelfa nd–Gelfand and Rocha-Caridi)\, and end by explaining that in the higher order cases\, (a) the category $\\mathcal{O}^\\mathcal{H}$ still has enoug h projectives and injectives\; (b) BGG reciprocity does not always hold "o n the nose"\, yet (c) the "Cartan matrix" (of simple multiplicities in pro jective covers) is still symmetric over $\\mathfrak{g} = \\mathfrak{sl}_2^ {\\oplus n}$.\n LOCATION:https://researchseminars.org/talk/ARCSIN/21/ END:VEVENT BEGIN:VEVENT SUMMARY:Apoorva Khare (Indian Institute of Science) DTSTART:20220609T060000Z DTEND:20220609T070000Z DTSTAMP:20260422T212604Z UID:ARCSIN/22 DESCRIPTION:Title: Higher order Verma modules\, and a positive formula for all highest weight modules - talk 3\nby Apoorva Khare (Indian Institute of Science) as p art of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric fu nctions in INdia\n\n\nAbstract\nIn this final talk\, we continue the study of higher order Verma modules and the higher order category $\\mathcal{O} ^\\mathcal{H}$ over a Kac–Moody algebra $\\mathfrak{g}$ (one may assume this to be $\\mathfrak{sl}_n$ throughout the talk\, without sacrificing no velty). After recalling the definitions\, we explain how BGG reciprocity f ails to hold "on the nose"\, yet does hold in a modified form over $\\math frak{g} = \\mathfrak{sl}_2^{\\oplus n}$. We then explain BGG resolutions a nd Weyl-Kac type character formulas\, for these modules in certain cases. (Joint with G.V.K. Teja.)\n LOCATION:https://researchseminars.org/talk/ARCSIN/22/ END:VEVENT END:VCALENDAR