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THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
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July
22-24, 2015
Workshop
on Recent Developments in the Geometry and Combinatorics of
Hessenberg Varieties
to
be held at The Fields Institute
Organizing
Committee: Hiraku Abe, Osaka City University Advanced
Mathematical Institute and University of Toronto
Megumi Harada, McMaster University
Julianna Tymoczko, Smith College
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Overview
Hessenberg varieties are subvarieties of a full flag variety which arise
in many areas of mathematics, including geometric representation theory,
numerical analysis, mathematical physics,
combinatorics, and algebraic geometry. Special families of Hessenberg varieties
include Springer varieties (which play a fundamental role in geometric representation
theory), Peterson varieties (which arise when we study the quantum cohomology
of flag varieties), and the toric varieties associated with Weyl chambers
(which provides a connection between root systems and toric varieties).
Exciting recent developments in this area include: (1) new connections
with graph theory, in connection with a conjecture of Stanley and Stembridge,
(2) the theory of ``generalized splines'' being pioneered by Tymoczko and
collaborators, used to study questions related to Hessenberg varieties and
Schubert calculus, and (3) the link to the theory of Newton-Okounkov bodies,
and their associated integrable systems. This workshop will provide researchers
an informal setting in which to exchange ideas and formulate open questions
at the forefront of the study of Hessenberg varieties.
Registration
and Funding Information
Registration fees for the workshop are $45/person (expenses cover coffee
and refreshment breaks).
Online registration has now closed. To register on site, please visit the
Fields reception desk before the start of the Workshop on Wednesday, July
22.
Schedule
| Wednesday,
July 22 |
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9:00
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Registration |
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9:30-10:30
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Martha Precup, Baylor
University: Affine pavings of Hessenberg varieties |
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10:30-11:00
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Coffee break |
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11:00-12:00
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Alex Yong, University
of Illinois at Urbana-Champaign: Combinatorial commutative
algebra and varieties in the flag manifold |
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12:00-2:00
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Group introduction session
(with catered lunch provided) |
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2:00-3:00
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Elizabeth Drellich,
University of North Texas: Equivariant Cohomology
of Peterson Varieties |
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3:00-3:30
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Coffee break |
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3:30-4:30
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Erik Insko, Florida
Gulf Coast University: Schubert calculus and the
intersection theory of the Peterson variety |
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4:45-5:15
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Tatsuya Horiguchi, Osaka
City University: The equivariant cohomology
rings of regular nilpotent Hessenberg varieties in Lie type A |
| Thursday,
July 23 |
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09:30-10:30
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John Shareshian,
Washington University: Regular semisimple Hessenberg
varieties and chromatic quasisymmetric functions |
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10:30-11:00
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Coffee break |
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11:00-12:00
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Hiraku Abe, University
of Toronto and Osaka City University Advanced Mathematics Institute:
Nilpotent vs. semisimple via representations of symmetric
groups |
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12:00-2:00
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Poster blitz and poster
session (with catered lunch provided) |
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2:00-3:00
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Julianna Tymoczko, Smith
College: Generalized splines and Hessenberg varieties |
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3:00-3:30
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Coffee break |
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3:30-4:00
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Nealy Bowden, University
of New Hampshire: Generalized splines over $\mathbb{Z}$
and $\mathbb{Z}/m\mathbb{Z}$ (part I) |
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4:15-4:45
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McCleary Philbin, Smith
College: Generalized splines over $\mathbb{Z}$
and $\mathbb{Z}/m\mathbb{Z}$ (part II) |
Open questions
and problems from the Workshop (pdf file)
Abstracts
Martha Precup (Baylor University)
Affine pavings of Hessenberg varieties
In this talk we will show that Hessenberg varieties corresponding to
nilpotent elements which are regular in a Levi factor are paved by affines.
The methods we will discuss generalize those used to show Springer fibers
are paved by affines originally due to De Concini, Lusztig, and Procesi.
We then provide a partial reduction from paving Hessenberg varieties for
arbitrary elements to paving those corresponding to nilpotent elements,
generalizing results of Tymoczko.
Alexander Yong (University of Illinois at Urbana-Champaign)
Combinatorial commutative algebra and varieties in the flag manifold
I'll discuss the analysis of some ``natural'' varieties sitting in the
flag manifold from the perspective of combinatorial commutative algebra.
The main case of the talk will be the Peterson variety (of type A); we
detail joint work with E. Insko. I'll also mention similar studies of
Schubert varieties, Richardson varieties and K-orbit closures, including
work with various subsets of A. Knutson, B. Wyser and A. Woo.
Elizabeth Drellich (University of North Texas)
Equivariant Cohomology of Peterson Varieties
The Peterson variety is the best understood of the regular nilpotent
Hessenberg varieties. Using a generic one dimensional torus action, we
can explicitly describe the equivariant cohomology of the Peterson variety.
This equivariant cohomology ring inherits a lot of its structure from
the full flag variety. In this talk we will see a module basis for cohomology
ring and Giambelli's formula for expressing each basis class in terms
of the ring generators.
Erik Insko (Florida Gulf Coast University)
Schubert calculus and the intersection theory of the Peterson variety
We will discuss the correlation between the affine pavings indexed by
torus fixed points of the Peterson variety and the algebraic combinatorics
of the symmetric group. This correlation gives us an elementary proof
that homology of the Peterson variety injects into the homology of the
flag variety. The proof counts the points of intersection between certain
Schubert varieties in the full flag variety and the Peterson variety to
(partially) expand the Peterson-Schubert classes in terms of flag variety's
basis of Schubert classes. We will also discuss several open questions
regarding expansions of Peterson-Schubert classes.
This talk is based on the recently published articles:
1. E. Insko, Schubert calculus and the homology of the Peterson variety,
Electronic Journal of Combinatorics, Volume 22, Issue 2, P2.26 (2015).
http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i2p26
2. E. Insko and J. Tymoczko, Affine pavings of regular nilpotent Hessenberg
varieties and intersection theory of the Peterson variety, Geometriae
Dedicata, (DOI) 10.1007/s10711-015-0093-5.
http://www.springer.com/home?SGWID=0-0-1003-0-0&aqId=2867045&download=1&checkval=bd68d95f12eea760a6de734f6a81dba7
(pdf download).
Tatsuya Horiguchi (Osaka City University)
The equivariant cohomology rings of regular nilpotent Hessenberg varieties
in Lie type A
Let $n$ be a fixed positive integer and $h : \{1,2,...,n\} \rightarrow
\{1,2,...,n\}$ a Hessenberg function. The main result is to give a systematic
method for producing an explicit presentation by generators and relations
of the equivariant and ordinary cohomology rings with $\mathbb{Q}$ coefficients
of any regular nilpotent Hessenberg variety in Lie type A. Specifically,
we give an explicit algorithm, depending only on the Hessenberg function
$h$, which produces the n defining relations in the equivariant cohomology
ring. Our result generalizes known results: for the case $h = (2,3,4,...,n,n)$,
which corresponds to the Peterson variety $Pet$, we recover the presentation
of the equivariant and ordinary cohomology ring of $Pet$ given previously
by Fukukawa, Harada, and Masuda. Moreover, in the case $h = (n,n,...,n)$,
for which the corresponding regular nilpotent Hessenberg variety is the
full flag variety $Flags(\mathbb{C}^n)$, we can explicitly relate the
generators of our ideal with those in the usual Borel presentation of
the cohomology ring of $Flags(\mathbb{C}^n)$. This is a joint work with
Hiraku Abe, Megumi Harada, and Mikiya Masuda.
John Shareshian (Washington University)
Regular semisimple Hessenberg varieties and chromatic quasisymmetric
functions
In joint work with Michelle Wachs (University of Miami), we defined a
refinement of Richard Stanley's chromatic symmetric function. Given a
(simple, loopless, finite) graph $G$, the chromatic symmetric function
$X_G(x_1,x_2,...)$ is a generating function describing all proper colorings
of $G$ with the positive integers. Our refinement $X_G(t,x_1,x_2,...)$
is defined on graphs with vertex set $\{1,2,...,n\}$ for any positive
integer $n$. In general, this is not a symmetric function. However, for
certain graphs naturally associated to type A Hessenberg varieties, this
refinement is indeed a symmetric function. We conjecture that in this
case, $X_G(t,x_1,x_2,...)$ is (up to a sign twist) the Frobeniusc haracteristic
of the representation of $S_n$ on the cohomology of the associated regular
semisimple Hessenberg variety, determined by the action on the moment
graph. After defining the relevant objects, I will explain how we came
to this conjecture, describe the evidence in its favor, and discuss connections
with a conjecture of Stanley and John Stembridge.
Hiraku Abe (University of Toronto and Osaka City
University Advanced Mathematics Institute)
Nilpotent vs. semisimple via representations of symmetric groups
In this talk, we will investigate the representations of symmetric groups
on the cohomology of regular semisimple Hessenberg varieties. Via this
representation, we will discuss a relation between cohomology rings of
``regular nilpotent" and ``regular semisimple" Hessenberg varieties
as a bridge connecting those two classes of Hessenberg varieties.
We will also consider the representations of symmetric groups on the
equivariant cohomology of regular semisimple Hessenberg varieties, and
we will explain how they behave when we vary the Hessenberg functions,
as a work in progress.
This is a joint work with Megumi Harada, Tatsuya Horiguchi, and Mikiya
Masuda.
Julianna Tymoczko (Smith College)
Generalized splines and Hessenberg varieties
Splines are a kind of smooth, piecewise-polynomial approximation originally
developed for engineering applications but now used widely in computer
graphics, data interpolation, and other fields. Billera and others pioneered
an algebraic approach to splines, using sophisticated techniques from
commutative and homological algebra. Independently, geometers and topologists
developed a construction of equivariant cohomology rings for large classes
of varieties that turns out to coincide with the ring of splines.
In this talk, we describe how to generalize the construction of splines
to a more natural algebraic and combinatorial setting, starting from a
commutative ring and a graph $G$. We'll also show how powerful this construction
can be: how the ring of splines naturally decomposes in terms of splines
for certain subgraphs of $G$, and how to construct generating sets for
the generalized splines. We will highlight examples from geometry (including
from Hessenberg varieties and related varieties) and give many open questions.
Nealy Bowden (University of New Hampshire)
and McCleary Philbin (Smith College)
Generalized splines over $\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$
Fix a ring $R$ and label the edges of a connected graph $G$ with ideals
in $R$. A generalized spline on $G$ over $R$ is a set of vertex labels
in $R$ that satisfy the condition that whenever two vertices are joined
by an edge, the labels on those vertices differ by a multiple of the label
on the edge by which they are joined. The collection of all splines on
$G$ forms a module whose properties vary significantly depending on our
choice of $R$. In this talk we survey recent work on generalized splines
in two closely related cases: when the ring is $\mathbb{Z}$ and when it
is $\mathbb{Z}/m\mathbb{Z}$.
We highlight the major similarities and differences between these two
cases, particularly when we look at the minimum generating sets of modules
of splines over $\mathbb{Z}$ and $\mathbb{Z}/m\mathbb{Z}$. We introduce
an algorithm that allows us to completely classify splines over $\mathbb{Z}/m\mathbb{Z}$
when $G$ is a connected graph. We then underscore the important connection
between splines over these two different rings by presenting an extension
of this algorithm that produces bases for splines over $\mathbb{Z}$. Finally
we discuss the multiplicative structure of the ring of splines over $\mathbb{Z}/m\mathbb{Z}$
and present multiplication tables for minimum generating set elements.
Aba Mbirika (University of Wisconsin-Eau Claire)
A co-FI-module structure on Springer representations
A sequence of $S_n$-representations $\{V_n\}$ is said to be uniformly
representation stable if the decomposition of $V_n = \bigoplus_{\lambda}
c_{\lambda,n} V(\lambda)_n$ into irreducible representations can be described
independently of $n$ for each $\lambda$---that is, the multiplicities
$c_{\lambda,n}$ are eventually independent of $n$ for each $\lambda$.
It is known that uniform representation stability holds for the cohomology
of flag varieties (the so-called diagonal coinvariant algebra), a well-known
Springer representation. But does it hold for all Springer representations?
In this talk we explore this question. Central to this exploration is
the co-FI-module structure (in the sense of Church-Ellenberg-Farb) of
a sequence of cohomology rings of Springer varieties parametrized by partitions
of $n$.
This is a joint work in progress with Julianna Tymoczko.
Jihyeon Jessie Yang (McMaster University)
Newton-Okounkov body theory and connection to Representation theory
A Newton-Okounkov body is a certain convex set in an Euclidean space,
which can be constructed from an object of interest. For example, when
this object is a projective toric variety, we can recover the very well-known
connections between toric geometry and polytopal geometry via Newton polytopes.
In this talk, we begin with the overview on the Newton-Okounkov body theory
in many different points of view depending on what the object of interest
is. Then we focus on the connections to Representation theory by considering
reductive algebraic groups.
Lauren Dedieu (McMaster University)
Newton-Okounkov Bodies of Peterson and Bott-Samelson Varieties
The theory of Newton-Okounkov bodies can be viewed as a generalization
of the theory of toric varieties; it associates a convex body to an arbitrary
variety (equipped with auxiliary data). Although initial steps have been
taken for formulating geometric situations under which the Newton-Okounkov
body is a rational polytope, there is much that is still unknown.
In particular, very few concrete and explicit examples have been computed
thus far. In this talk, I will discuss explicitly computations of Newton-Okounkov
bodies of Peterson and Bott-Samelson varieties (for certain classes of
auxiliary data on these varieties). Both of these varieties arise, for
instance, in the geometric study of representation theory. This is work
in progress and will be part of my Ph.D. thesis.
Participants
Hiraku Abe, Osaka City University / University of Toronto
Nealy Bowden, University of New Hampshire
Peter Crooks, University of Toronto
Lauren DeDieu, McMaster University
Elizabeth Drellich, University of North Texas
Megumi Harada, McMaster University
Tatsuya Horiguchi, Osaka City University
Erik Insko, Florida Gulf Coast University
Lisa Jeffrey, University of Toronto
Yael Karshon, University of Toronto (UTM)
Kiumars Kaveh, University of Pittsburgh
Jeremy Lane, University of Toronto
Christopher Manon, George Mason University
Aba Mbirika, University of Wisconsin-Eau Claire
McCleary Philbin, Smith College
Martha Precup, Baylor University
Paul Selick, University of Toronto
John Shareshian, Washington University
Julianna Tymoczko, Smith College
Michelle Wachs, University of Miami
Jihyeon Jessie Yang, McMaster University
Alexander Yong, University of Illinois at Urbana Champaign
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