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BEGIN:VEVENT
SUMMARY:Pritam Majumder (Tata Institute of Fundamental Research\, Mumbai)
DTSTART:20210219T093000Z
DTEND:20210219T103000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/1
DESCRIPTION:Title: On characterizing line graphs of hypergraphs\nby Pritam Majumder
(Tata Institute of Fundamental Research\, Mumbai) as part of ARCSIN - Alg
ebra\, Representations\, Combinatorics and Symmetric functions in INdia\n\
n\nAbstract\nA hypergraph is given by a finite set of vertices together wi
th a collection of its subsets\, called edges\, of that set. A hypergraph
is called 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
vertices bijectively correspond to the edges of the hypergraph and there i
s an edge between two vertices of the line graph if the corresponding edge
s in the hypergraph have non zero intersection. The characterization of li
ne graphs of 2-uniform hypergraphs (graphs) have been extensively studied.
The characterization of line graphs of k-uniform hypergraphs for k>2 is p
oorly understood. Partial results exist for linear hypergraphs (intersecti
on of any two edges is at most 1 vertex). We study the problem of \ncharac
terizing line graphs of k-uniform hypergraphs with bounded pair-degree by
a finite class of forbidden subgraphs. We show that such a characterizatio
n is possible if we consider line graphs with certain minimum edge-degree.
Time permitting\, we shall discuss about some other reconstruction proble
ms for hypergraphs\, namely\, characterizing degree sequences of hypergrap
hs and characterizing face numbers (f-vectors) of simplicial complexes.\n
LOCATION:https://stable.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:20260404T110831Z
UID:ARCSIN/2
DESCRIPTION:Title: Multiplicity of trivial and sign representations of $S_n$ in hook-sh
aped representations of $GL_n$.\nby Sridhar P Naryanan (The Institute
of Mathematical Sciences\, Chennai) as part of ARCSIN - Algebra\, Represen
tations\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nLe
t $W_\\lambda$ be an irreducible representation of $GL_n$ (for\npartition
$\\lambda$ with $\\leq n$ parts). Let $V_\\mu$ be an irreducible\nrepresen
tation of $S_n$ (for partition $\\mu \\vdash n$). Then $$W_\\lambda=\n\\su
m_{\\mu \\vdash n} r_{\\lambda \\mu} V_\\mu.$$\nThe coefficients $r_{\\lam
bda\\mu}$ are the restriction coefficients. The\nrestriction problem is to
find combinatorial objects that these coefficients\ncount. We find such o
bjects when $\\lambda$ is the hook shape and $\\mu=(n)$\nor $\\mu= (1^n)$
using the theory of character polynomials and a simple\nsign-reversing inv
olution.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/2/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Souvik Pal (Harish Chandra Research Institute\, Allahabad)
DTSTART:20210326T093000Z
DTEND:20210326T103000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/3
DESCRIPTION:Title: Level zero integrable modules with finite-dimensional weight spaces
for the graded Lie tori\nby Souvik Pal (Harish Chandra Research Insti
tute\, Allahabad) as part of ARCSIN - Algebra\, Representations\, Combinat
orics and Symmetric functions in INdia\n\n\nAbstract\nAn important problem
in the representation theory of affine and toroidal Lie algebras is to cl
assify all possible irreducible integrable modules with finite-dimensional
weight spaces. The centres of both affine and toroidal Lie algebras are s
panned by finitely many elements. If all these central elements act trivia
lly on a module\, we say that the representation has level zero\, otherwis
e it is said to have non-zero level. The classification of these irreducib
le integrable modules with finite-dimensional weight spaces over the affin
e Kac-Moody algebras (both twisted and untwisted) have been completely set
tled by V. Chari and A. Pressley. This was subsequently generalized by S.
Eswara Rao for the (untwisted) toroidal Lie algebras. Recently\, the afore
mentioned irreducible integrable modules of non-zero level have been class
ified for a more general class of Lie algebras\, namely the graded Lie tor
i\, which are multivariable generalizations of twisted affine Kac-Moody al
gebras. In this talk\, I shall address the mutually exclusive problem and
henceforth classify all the level zero irreducible integrable modules with
finite-dimensional weight spaces for this graded Lie tori.\n
LOCATION:https://stable.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:20260404T110831Z
UID:ARCSIN/4
DESCRIPTION:Title: Representations of Compact Special Linear Groups of Degree Two\n
by Hassain Maliyekkal (The Institute of Mathematical Sciences\, Chennai\,
India) as part of ARCSIN - Algebra\, Representations\, Combinatorics and S
ymmetric functions in INdia\n\n\nAbstract\nLet $\\mathcal{O}$ be the ring
of integers of a non-Archimedean local field and $\\wp$ be its maximal ide
al. In this talk\, our focus is on the construction of the continuous comp
lex irreducible representations of the group $\\mathrm{SL}_2(\\mathcal{O})
$ and to describe their representation growth. We will also discuss some r
esults about group algebras of $\\mathrm{SL}_2(\\mathcal{O}/{\\wp}^r)$ for
large $r$.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/4/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Digjoy Paul (Tata Institute of Fundamental Research)
DTSTART:20211028T060000Z
DTEND:20211028T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/5
DESCRIPTION:Title: Symmetric $q\,t$ Catalan polynomials\nby Digjoy Paul (Tata Insti
tute of Fundamental Research) as part of ARCSIN - Algebra\, Representation
s\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nThe $q\,
t$-Catalan functions $C_n(q\,t)$\, an $q\, t$- analogue of Catalan number
s\, were first introduced in connection with Macdonald polynomials and Gar
sia–Haiman’s theory of diagonal harmonics [1996] as certain rational f
unctions in $q$ and $t$. Haglund [2003] and shortly after that\, Haiman an
nounced two combinatorial interpretations of $C_n(q\,t)$ as a weighted sum
over all Dyck paths. An open problem related to these polynomials is a co
mbinatorial proof of its symmetry in $q$ and $t$.\n\nWe define two symmetr
ic $q\,t$ Catalan polynomials on Dyck paths and provide proof of the symme
try by establishing an involution on plane trees. We also give a combinato
rial proof of a result by Garsia et al. regarding parking functions and th
e number of connected graphs. This is joint work with Joseph Pappe and Ann
e Schilling.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/5/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Manjunath Krishnapur (Indian Institute of Science\, Bangalore)
DTSTART:20211111T060000Z
DTEND:20211111T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/6
DESCRIPTION:Title: Free probability\, free convolution and associated combinatorics of
non-crossing partitions\nby Manjunath Krishnapur (Indian Institute of
Science\, Bangalore) as part of ARCSIN - Algebra\, Representations\, Combi
natorics and Symmetric functions in INdia\n\n\nAbstract\nThis is an exposi
tory talk on free probability\, which is a part of operator theory with ve
ry strong parallels to probability theory. In particular\, we shall focus
on free convolution\, which is a binary operation on measures on the real
line that is different from the usual convolution that arises when one add
s independent random variables. Getting rid of analysis and expressing eve
rything algebraically\, the difference between the two forms of convolutio
n arises from the difference between the lattice of all set partitions of
a finite set and the lattice of all non-crossing set partitions of the sam
e. We would also like to explain how free convolution arises in random mat
rix theory (what are the eigenvalues of the sum of two large matrices?) an
d in asymptotic representation theory of symmetric groups (what are the Li
ttlewood-Richardson coefficients of large partitions?). \n\nNot much know
ledge of probability will be assumed. We shall refer to Bernoulli and Gaus
sian random variables\, independence and convolution\, central limit theor
em\, mainly to explain the analogies. The talk should be accessible to adv
anced undergraduates.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/6/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Bishal Deb (University College London)
DTSTART:20211118T060000Z
DTEND:20211118T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/7
DESCRIPTION:Title: Analysing a strategy for a card guessing game via continuously incre
asing subsequences in multiset permutations\nby Bishal Deb (University
College London) as part of ARCSIN - Algebra\, Representations\, Combinato
rics and Symmetric functions in INdia\n\n\nAbstract\nConsider the followin
g card guessing game introduced by Diaconis and Graham (1981): there is a
shuffled deck of $mn$ cards with $n$ distinct cards numbered $1$ to $n$\,
each appearing with multiplicity $m$. In each round\, the player has to gu
ess the top card of the deck\, and is then told whether the guess was corr
ect or not\, the top card is then discarded and then the game continues wi
th the next card. This is known as the partial feedback model. The aim is
to maximise the number of correct guesses. One possible strategy is the sh
ifting strategy in which the player keeps guessing $1$ every round until t
he guess is correct in some round\, and then keeps guessing $2$\, and then
$3$ and so on. We are interested in finding the expected score using this
strategy.\n\nWe can restate this problem as finding the expectation of th
e largest 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 this talk\, we show that this number is $m+1 - 1/(m+2)$ plus an expo
nential error term. This confirms a conjecture of Diaconis\, Graham\, He a
nd Spiro.\n\nThis talk will be at an interface between combinatorics\, pro
bability and analysis and will feature an unexpected appearance of the Tay
lor polynomials of the exponential function. This is based on joint work w
ith Alexander Clifton\, Yifeng Huang\, Sam Spiro and Semin Yoo.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/7/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nishu Kumari (Indian Institute of Science)
DTSTART:20211125T060000Z
DTEND:20211125T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/8
DESCRIPTION:Title: Factorization of Classical Characters twisted by Roots of Unity\
nby Nishu Kumari (Indian Institute of Science) as part of ARCSIN - Algebra
\, Representations\, Combinatorics and Symmetric functions in INdia\n\n\nA
bstract\nIn representation theory\, Schur polynomials are the characters o
f the irreducible polynomial representations of the classical groups of ty
pe A\, namely $GL_n(\\mathbb{C})$. \nMotivated by a celebrated result of K
ostant\, D. Prasad considered factorization of Schur polynomials \nin $tn$
variables\, for $t \\geq 2$\, a fixed positive integer\, \nspecialized to
$(\\exp(2 \\pi \\iota k/t) x_j)_{0 \\leq k \\leq t-1\, 1 \\leq j \\leq n}
$ (Israel J. Math.\, 2016). He characterized partitions for which these Sc
hur polynomials are nonzero and showed that if the Schur polynomial is non
zero\, it factorizes into characters of smaller classical groups of Type A
.\n\nWe generalize Prasad's result to the irreducible characters of classi
cal groups \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 specializa
tion. \nWe give a uniform approach for all cases. \nThe proof uses Cauchy-
type determinant formulas for these characters and involves a careful stud
y of the beta sets of partitions. This is joint work with Arvind Ayyer and
is available at https://arxiv.org/abs/2109.11310.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/8/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hemanshu Kaul (Illinois Institute of Technology)
DTSTART:20220106T050000Z
DTEND:20220106T060000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/9
DESCRIPTION:Title: Chromatic polynomial and counting DP-colorings of graphs : Problems
and progress\nby Hemanshu Kaul (Illinois Institute of Technology) as p
art of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric fu
nctions in INdia\n\n\nAbstract\nIn 1912\, Birkhoff\, introduced the chrom
atic polynomial 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 natural generalization of ordinary coloring where each vertex has a res
tricted 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 b
een studied since 1990.\n\nDP-coloring (also called correspondence colorin
g) is a generalization of list coloring that has been widely studied in re
cent years after its introduction by Dvorak and Postle in 2015. Intuitivel
y\, DP-coloring is a variation on list coloring where each vertex in the g
raph still gets a list of colors\, but identification of which colors are
different can change from edge to edge. It is equivalent to the question o
f finding independent transversals in a (DP-)cover of a graph. In this tal
k\, we will introduce a DP-coloring analogue of the chromatic polynomial c
alled the DP color function\, ask several fundamental open questions about
it\, and give an overview of the progress made on them. We show that whil
e the DP color function behaves similar to the list color function and chr
omatic polynomial for some graphs\, there are also some surprising fundame
ntal differences. \n\nThe results are based on joint work with Jeffrey Mud
rock (CLC)\, as well as several groups of students.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/9/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Alexander Yong (UIUC)
DTSTART:20220113T123000Z
DTEND:20220113T133000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/10
DESCRIPTION:Title: Newell-Littlewood numbers\nby Alexander Yong (UIUC) as part of
ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric functions
in INdia\n\n\nAbstract\nThe Newell-Littlewood numbers are defined in term
s of the \nLittlewood-Richardson coefficients from algebraic combina
torics. Both \nappear in representation theory as tensor product multiplic
ities for a\nclassical Lie group. This talk concerns the question: \n\n
Which multiplicities are nonzero? \n\nIn 1998\, Klyachko establ
ished common linear inequalities defining \nboth the eigencone for sums of
Hermitian matrices and the saturated \nLittlewood-Richardson cone. We pro
ve some analogues of Klyachko's nonvanishing\nresults for the Newell-Littl
ewood numbers.\n\nThis is joint work with Shiliang Gao (UIUC)\, Gidon Orel
owitz (UIUC)\, and\nNicolas Ressayre (Universite Claude Bernard Lyon I). T
he presentation is based on\narXiv:2005.09012\, arXiv:2009.09904\, and arX
iv:2107.03152.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/10/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Sunil Chebolu (Illinois State University)
DTSTART:20220127T060000Z
DTEND:20220127T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/12
DESCRIPTION:Title: How many units can a commutative ring have?\nby Sunil Chebolu (
Illinois State University) as part of ARCSIN - Algebra\, Representations\,
Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nLaszlo Fuch
s posed the following problem in 1960\, which remains open: classify the a
belian groups occurring as the group of all units in a commutative ring. I
n this talk\, I will provide an elementary solution to a simpler\, related
problem: find all cardinal numbers occurring as the cardinality of the gr
oup of all units in a commutative ring. As a by-product\, we obtain a solu
tion to Fuchs' problem for the class of finite abelian p-groups when p is
an odd prime.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/12/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Samrith Ram (IIIT Delhi)
DTSTART:20220203T083000Z
DTEND:20220203T093000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/13
DESCRIPTION:Title: Set Partitions\, Tableaux\, and Subspace Profiles under Regular Spl
it Semisimple Matrices\nby Samrith Ram (IIIT Delhi) as part of ARCSIN
- Algebra\, Representations\, Combinatorics and Symmetric functions in INd
ia\n\n\nAbstract\nIn this talk we will introduce a family of polynomials $
b_\\lambda(q)$ indexed by integer partitions $\\lambda$. These polynomials
arise from an intriguing connection between two classical combinatorial c
lasses\, namely set partitions and standard tableaux. The polynomials $b_\
\lambda(q)$ can be derived from a new statistic on set partitions called t
he interlacing number which is a variant of the well-known crossing number
of a set partition. These polynomials also have several interesting speci
alizations: $b_\\lambda(1)$ enumerates the number of set partitions of sh
ape $\\lambda$ and $b_\\lambda(0)$ counts the number of standard tableaux
of shape $\\lambda$ while $b_\\lambda(-1)$ equals the number of standard s
hifted tableaux of shape $\\lambda$ respectively. When $q$ is a prime powe
r $b_\\lambda(q)$ counts (up to factors of $q$ and $q-1$) the number of su
bspaces in a finite vector space that transform under a regular diagonal m
atrix in a specified manner.\n\nThis is joint work with Amritanshu Prasad.
\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/13/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Santosh Nadimpalli (IIT Kanpur)
DTSTART:20220217T060000Z
DTEND:20220217T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/14
DESCRIPTION:Title: On uniqueness of branching to fixed point Lie subalgebras\nby S
antosh Nadimpalli (IIT Kanpur) as part of ARCSIN - Algebra\, Representatio
ns\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nLet $\\
mathfrak{g}$ be a complex semisimple Lie algebra and let\n $\\theta$ be a
finite order automorphism of $\\mathfrak{g}$. We assume\n that any ${\\r
m A}_{2n}$-type $\\theta$-stable indecomposable ideal of\n $\\mathfrak{g}
$ is simple and any ${\\rm D}_k$\, ${\\rm A}_{2k+1}$ and\n ${\\rm E}_6$-t
ype $\\theta$-stable indecomposable ideal of\n $\\mathfrak{g}$ has length
at most $2$. Let $\\mathfrak{g}_0$ be the\n fixed point subalgebra of $\
\mathfrak{g}$. In this talk\, for any\n irreducible finite dimensional r
epresentations $V_1$ and $V_2$ of\n $\\mathfrak{g}$\, we show that\n${\\r
m res}_{\\mathfrak{g}_0}V_1\\simeq \n{\\rm res}_{\\mathfrak{g}_0}V_2$ if a
nd only if $V_2$ is isomorphic to\n$V_1^\\sigma$\, for some outer automorp
hism $\\sigma$ of $\\mathfrak{g}$.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/14/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Shraddha Srivastava (Uppsala University)
DTSTART:20220505T060000Z
DTEND:20220505T070000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/15
DESCRIPTION:Title: Diagram categories and reduced Kronecker coefficients\nby Shrad
dha Srivastava (Uppsala University) as part of ARCSIN - Algebra\, Represen
tations\, Combinatorics and Symmetric functions in INdia\n\n\nAbstract\nPa
rtition 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 partit
ion category admits a triangular decomposition similar to a triangular dec
omposition of the universal enveloping algebra of a finite dimensional com
plex semisimple Lie algebra. In such a decomposition\, the direct sum of s
ymmetric group algebras plays a role analogous to Cartan subalgebra and th
is provides a natural approach to the representation theory of the partiti
on category. The tensor structure on the partition category induces a ring
structure on the associated Grothendieck group. Reduced Kronecker coeffic
ients for symmetric groups appear as structure constants in the Grothendie
ck ring. \n\nIn this talk\, we discuss the partition category and its conn
ection to reduced Kronecker coefficients (these are results of several aut
hors). We introduce the multiparameter colored partition category where th
e Cartan subalgebra in the corresponding triangular decomposition is given
by complex reflection groups of type $G(r\,1\,n)$. The multiparameter col
ored partition category also admits a tensor structure. If time permits\,
we also relate the associated Grothendieck ring for this category with the
ring of symmetric functions. This talk is based on joint work with Volody
myr Mazorchuk.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/15/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Ashish Mishra (Universidade Federal do Pará\, Belém)
DTSTART:20220303T100000Z
DTEND:20220303T110000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/16
DESCRIPTION:Title: The Jucys--Murphy elements\nby Ashish Mishra (Universidade Fede
ral do Pará\, Belém) as part of ARCSIN - Algebra\, Representations\, Com
binatorics and Symmetric functions in INdia\n\n\nAbstract\nThe representat
ion theory of a multiplicity free tower of finite-dimensional semisimple a
ssociative algebras is determined by the actions of Jucys--Murphy elements
. These elements were discovered independently by Jucys and Murphy for th
e symmetric groups\, and later on\, these elements played an important ro
le in the development of spectral approach to the representation theory of
symmetric groups given by Okounkov and Vershik. The motivation for the sp
ectral approach comes from the work of Gelfand and Tsetlin on the irreduci
ble finite-dimensional modules of general linear Lie algebras. \n\nAfter a
brief description of the history and fundamental properties of Jucys--Mur
phy elements\, our main objective in this seminar is to describe these ele
ments and to study their applications in the representation theory of foll
owing algebras: (i) partition algebras for complex reflection groups\, (ii
) rook monoid algebras\, and (iii) totally propagating partition algebras.
The results presented in this seminar are joint work with Dr. Shraddha Sr
ivastava.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/16/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Hugh r Thomas (Université du Québec à Montréal)
DTSTART:20220428T123000Z
DTEND:20220428T133000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/17
DESCRIPTION:Title: The Robinson--Schensted--Knuth correspondence via quiver representa
tions\nby Hugh r Thomas (Université du Québec à Montréal) as part
of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetric functi
ons in INdia\n\n\nAbstract\nThe RSK correspondence is a multi-faceted jewe
l at the heart of algebraic combinatorics. In one of its incarnations\, it
is a piecewise-linear bijection between an orthant and a much more compli
cated cone 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 co
nstruct it which arises naturally out of the theory of representations of
quivers. No knowledge of quiver representations will be assumed. This talk
is based on joint work with Al Garver and Becky Patrias.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/17/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Arvind Ayyer (Indian Institute of Science\, Bangalore)
DTSTART:20220311T083000Z
DTEND:20220311T093000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/18
DESCRIPTION:Title: A multispecies totally asymmetric zero range process and Macdonald
polynomials\nby Arvind Ayyer (Indian Institute of Science\, Bangalore)
as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symmetr
ic functions in INdia\n\nLecture held in Alladi Ramakrishnan Hall\, IMSc\,
Chennai.\n\nAbstract\nMacdonald polynomials are a remarkable family of sy
mmetric functions that\nare known to have connections to combinatorics\, a
lgebraic geometry and\nrepresentation theory. Due to work of Corteel\, Man
delshtam and Williams\, it\nis known that they are related to the asymmetr
ic simple exclusion process\n(ASEP) on a ring.\n\nThe modified Macdonald p
olynomials are obtained from the Macdonald\npolynomials using an operation
called plethysm. It is natural to ask whether\nthe modified Macdonald pol
ynomials are related to some other particle\nsystem. In this talk\, we ans
wer this question in the affirmative via a\nmultispecies totally asymmetri
c zero-range process (TAZRP). We also present\na Markov process on tableau
x that projects to the TAZRP and derive formulas\nfor stationary probabili
ties and certain correlations. This is joint work\nwith Olya Mandelshtam a
nd James Martin.\n\nThis will be a hybrid event. Note that the usual Zoom
link for the seminar is not valid.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/18/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Nicolas Libedinsky (Universidad de Chile)
DTSTART:20220526T110000Z
DTEND:20220526T120000Z
DTSTAMP:20260404T110831Z
UID:ARCSIN/19
DESCRIPTION:Title: Introsurvey of representation theory\nby Nicolas Libedinsky (Un
iversidad de Chile) as part of ARCSIN - Algebra\, Representations\, Combin
atorics and Symmetric functions in INdia\n\n\nAbstract\nWe will give a sum
mary of the paper "Introsurvey of representation theory". We will start wi
th classical Schur-Weyl duality\, then introduce Iwahori-Hecke algebra and
Soergel bimodules. We will finish with some of the fascinating theorems a
nd conjectures around these objects.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/19/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (Indian Institute of Science)
DTSTART:20220520T060000Z
DTEND:20220520T070000Z
DTSTAMP:20260404T110831Z
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 Scienc
e) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symme
tric functions 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 begin with several positive weight-formulas for arbitrary non-integr
able simple modules\, and mention the equivalence of several "first order"
data that helps prove these formulas. We then discuss the notion of "high
er order holes" in the weights\, and use these to present two positive wei
ght-formulas for arbitrary modules $V$. One of these is in terms of "highe
r order Verma modules". (Joint with G.V.K. Teja and with Gurbir Dhillon.)\
n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/20/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (Indian Institute of Science)
DTSTART:20220602T060000Z
DTEND:20220602T070000Z
DTSTAMP:20260404T110831Z
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 Scienc
e) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symme
tric functions 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 begin by recalling the notation\, "higher order holes"\, and "higher
order Verma modules" (along with their universal property\, via examples)
. We then recall our result from last time: the weights of any highest wei
ght module equal the weights of its "higher order" Verma cover.\n\nNext\,
we define the higher order category $\\mathcal{O}^\\mathcal{H}$\, and reca
ll some properties in the zeroth and first order cases (work of Bernstein
–Gelfand–Gelfand and Rocha-Caridi)\, and end by explaining that in the
higher order cases\, (a) the category $\\mathcal{O}^\\mathcal{H}$ still h
as enough projectives and injectives\; (b) BGG reciprocity does not always
hold "on the nose"\, yet (c) the "Cartan matrix" (of simple multiplicitie
s in projective covers) is still symmetric over $\\mathfrak{g} = \\mathfra
k{sl}_2^{\\oplus n}$.\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/21/
END:VEVENT
BEGIN:VEVENT
SUMMARY:Apoorva Khare (Indian Institute of Science)
DTSTART:20220609T060000Z
DTEND:20220609T070000Z
DTSTAMP:20260404T110831Z
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 Scienc
e) as part of ARCSIN - Algebra\, Representations\, Combinatorics and Symme
tric functions in INdia\n\n\nAbstract\nIn this final talk\, we continue th
e study of higher order Verma modules and the higher order category $\\mat
hcal{O}^\\mathcal{H}$ over a Kac–Moody algebra $\\mathfrak{g}$ (one may
assume this to be $\\mathfrak{sl}_n$ throughout the talk\, without sacrifi
cing novelty). After recalling the definitions\, we explain how BGG recipr
ocity fails to hold "on the nose"\, yet does hold in a modified form over
$\\mathfrak{g} = \\mathfrak{sl}_2^{\\oplus n}$. We then explain BGG resolu
tions and Weyl-Kac type character formulas\, for these modules in certain
cases. (Joint with G.V.K. Teja.)\n
LOCATION:https://stable.researchseminars.org/talk/ARCSIN/22/
END:VEVENT
END:VCALENDAR