# Schedule for: 18w5130 - Hessenberg Varieties in Combinatorics, Geometry and Representation Theory

Beginning on Sunday, October 21 and ending Friday October 26, 2018

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, October 21 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 22:00 | Informal gathering (Corbett Hall Lounge (CH 2110)) |

Monday, October 22 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |

09:00 - 10:00 |
Hiraku Abe: An introduction to Hessenberg varieties ↓ In this talk, I will give a brief survey of recent developments on Hessenberg varieties. The goal of this talk is to share the idea that Hessenberg varieties can be studied from various perspectives. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 | Julianna Tymoczko: K-theory/cohomology classes of Hessenberg varieties (TCPL 201) |

11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:00 - 14:00 |
Guided Tour of The Banff Centre ↓ Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus. (Corbett Hall Lounge (CH 2110)) |

14:00 - 14:15 |
Group Photo ↓ Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo! (TCPL 201) |

14:15 - 15:15 |
Timothy Chow: A proof of the Shareshian-Wachs conjecture ↓ Motivated by a 1993 conjecture of Stanley and Stembridge,
Shareshian and Wachs conjectured that the characteristic map takes the
dot action of the symmetric group on the cohomology of a regular
semisimple Hessenberg variety to $\omega XG(t)$, where XG(t) is the
chromatic quasisymmetric function of the incomparability graph G of
the corresponding natural unit interval order, and $\omega$ is the
usual involution on symmetric functions. Our main result is a proof
of the Shareshian-Wachs conjecture. The proof makes essential use of
both geometric arguments and combinatorial arguments; the talk will
focus on the combinatorics, but we also briefly describe the geometric
ingredients. This is joint work with Patrick Brosnan. (TCPL 201) |

14:30 - 15:00 | Coffee Break (TCPL Foyer) |

15:45 - 16:45 |
Mathieu Guay-Paquet: Linear relations between q-chromatic symmetric functions ↓ The Brosnan--Chow--Shareshian--Wachs theorem gives a deep connection
between the cohomology of regular semisimple Hessenberg varieties on
one hand, and the $q$-chromatic symmetric functions of unit interval
orders ($q$-CSFs) from combinatorics.
Through this connection, a very interesting interplay of combinatorial
and geometric results and techniques becomes possible, leading to
progress on the long-standing $e$-positivity conjecture of
Stanley--Stembridge and its generalizations.
This talk is about recent combinatorial results on $q$-chromatic
symmetric functions: (1) a complete description of the linear
relations between $q$-CSFs; (2) a description of a large class of such
linear relations which express a $q$-CSF as a (TCPL 201) positive
combinations of other $q$-CSFs; and (3) a slight generalization of the
$e$-positivity conjecture suggested by these results. |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

Tuesday, October 23 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Martha Precup: The Betti numbers of Hessenberg varieties ↓ This talk will begin with a survey of results describing the Betti numbers of Hessenberg varieties. There are many geometric and combinatorial applications of these formulas. In the second half of the talk, I will report on recent joint work with M. Harada in which we prove an inductive formula for the Betti numbers of certain regular Hessenberg varieties called abelian Hessenberg varieties. Using a theorem of Brosnan and Chow, this formula yields a proof of the Stanley-Stembridge conjecture for this special case. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Erik Insko: Singularities of Hessenberg varieties ↓ Hessenberg varieties are subvarieties of the flag variety with important connections to representation theory, algebraic geometry, and combinatorics. The local geometric structure of Hessenberg varieties can often be studied using cell decompositions, group actions, patch ideals, and the combinatorics of the symmetric group. In this talk, we will give a survey of results regarding the singularities of Hessenberg varieties. This is based on joint works with Alex Yong and Martha Precup. (TCPL 202) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

13:15 - 14:15 |
Mikiya Masuda: The cohomology rings of regular semisimple Hessenberg varieties for $h=(h(1),n,\ldots,n)$ ↓ I will talk about an explicit ring presentation of the cohomology ring of the regular semisimple Hessenberg variety in the flag variety $Fl(\mathbb{C}^n)$ with Hessenberg function $h=(h(1),n,\dots,n)$ where $h(1)$ is an arbitrary integer between $2$ and $n$. This is joint work with Hiraku Abe and Tatsuya Horiguchi. (TCPL 201) |

14:30 - 15:30 |
Tatsuya Horiguchi: The cohomology rings of regular nilpotent Hessenberg varieties and Schubert polynomials ↓ A regular nilpotent Hessenberg variety Hess(N,h) is a subvariety of a flag variety determined by a Hessenberg function h. I will explain a relation between the cohomology ring of a regular nilpotent Hessenberg variety and Schubert polynomials. To describe an explicit presentation of the cohomology ring of a regular nilpotent Hessenberg variety, polynomials $f_{i,j}$ were introduced by Abe-Harada-Horiguchi-Masuda. In this talk, I will show that every polynomial $f_{i,j}$ is an alternating sum of certain Schubert polynomials $\mathfrak{S}_{w_k^{(i,j)}}$ (k=1,2,...,i-j). Moreover, we can interpret the permutations $w_k^{(i,j)}$ from a geometric viewpoint under the circumstances of having a codimension one regular nilpotent Hessenberg variety Hess(N,h′) in the original regular nilpotent Hessenberg variety Hess(N,h) where the Hessenberg function h' is obtained from the original Hessenberg function h and (i,j). (TCPL 201) |

15:15 - 15:45 | Coffee Break (TCPL Foyer) |

16:00 - 17:00 |
James Carrell: Cohomology algebras of varieties with a 'regular' $(\mathbb{G}_a,\mathbb{G}_m)$-action ↓ This is an expository talk on "regular $(\mathbb{G}_a,\mathbb{G}_m)$-actions" on projective varieties (over the complexes). A $(\mathbb{G}_a,\mathbb{G}_m)$-action is the same thing as an action of a Borel subgroup of $\mathrm{SL}(2)$.
We call a $(\mathbb{G}_a,\mathbb{G}_m)$-action regular when the maximal unipotent subgroup $\mathbb{G}_a$ has a unique fixed point. There are lots of examples of varieties with a regular $(\mathbb{G}_a,\mathbb{G}_m)$-action such as
Schubert varieties. Of particular interest are regular nilpotent Hessenberg varieties. The main point is that the cohomology algebra of a smooth projective variety X with $(\mathbb{G}_a,\mathbb{G}_m)$-action
is isomorphic with the fixed point scheme of the $\mathbb{G}_a$ action, and this is also true of many singular $(\mathbb{G}_a,\mathbb{G}_m)$-stable subvarieties such as the two classes mentioned above. The $\mathbb{G}_m$-equivariant cohomology admits a nice description which I will also explain. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Wednesday, October 24 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Andy Wilson: Macdonald polynomials and chromatic functions ↓ I will discuss how chromatic quasisymmetric functions can be viewed as modifications of LLT polynomials, certain building blocks from the combinatorial theory of Macdonald polynomials. I will also explain how Macdonald polynomials enter the picture, and what I hope can be gained from these relationships. Based on joint work with Jim Haglund. (TCPL 202) |

10:00 - 10:15 | Coffee Break (TCPL Foyer) |

10:15 - 10:45 |
Jim Haglund: The $q \to q+1$ phenomenon ↓ LLT polynomials depend on a tuple of skew shapes and a parameter q. They occur in formulas for Macdonald polynomials, where the skew shapes are ribbons, and in character formulas for Sn modules such as the diagonal coinvariant ring. LLT polynomials are known to be Schur positive, but combinaotrial formulas for the Schur coefficients are known only for special cases. Several researchers including Alexandersson, Bergeron, and Garsia have noticed independently that when q is replaced by q+1, LLT polynomials often become e-positive, and it has been conjectured that for tuples of vertical strips, this is always the case. In this talk we discuss some partial results on the e-expansion of LLT polynomials corresponding to Dyck paths which rise in the character of the diagonal coinvariant ring; in particular we discuss a conjecture for the coefficient of e_n in their e-expansion. (TCPL 202) |

11:00 - 12:00 |
Mark Skandera: Trace functions, the Kazhdan-Lusztig basis, and chromatic quasisymmetric functions ↓ When G is the indifference graph of a natural unit interval
order, the Shareshian-Wachs chromatic quasisymmetric function is
symmetric, and serves as a generating function for Hecke algebra trace
evaluations. In particular, every coefficient appearing in every
common symmetric function expansion is the evaluation of a common
Hecke algebra trace at a modified Kazhdan-Lusztig basis element
indexed by a 3412-, 4231- avoiding permutation. We will discuss
various combinatorial interpretations of these, and related ideas in
generating functions. (TCPL 202) |

12:00 - 13:30 | Lunch (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Thursday, October 25 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

09:00 - 10:00 |
Satoshi Murai: Hessenbergs and hyperplane arrangements part I ↓ In this talk, I will show that there is an interesting connection
between cohomology rings of regular nilpotent Hessenberg varieties and
logarithmic derivation modules of hyperplane arrangements. In
particular, I will explain that this connection gives an affirmative
answer to a conjecture of Sommers and Tymoczko on Poincare polynomials
of regular nilpotent Hessenberg varieties. If time allows, I also
explain how this connection can be applied to compute relations of
cohomology rings of regular nilpotent Hessenberg varieties explicitly. (TCPL 202) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Takuro Abe: Hessenbergs and hyperplane arrangements part II ↓ A hyperplane arrangement is a finite set of linear hyperplanes in the
complex vector space.
From each hyperplane arrangement and arbitrary degree two homogeneous
polynomial, we can define
a finite dimensional algebra, called the Solomon-Terao algebra. If we
choose an arrangement the
ideal arrangement corresponding to the reflecting hyperplanes belonging
to the lower ideal in the
root poset (which corresponds to a Hessenberg space), and the lowest
degree basic invariant polynomial
which is always degree two, then their Solomon-Terao algebra coincides
with the cohomology ring
of the regular nilpotent Hessenberg variety, as shown in the talk by
Murai.
We discuss the origin and properties of the Solomon-Terao algebra coming
from so called the mysterious
Solomon-Terao formula and polynomials by Luis Solomon and Hiroaki Terao
in 1986. We discuss also the
possibility whether it could be a cohomology ring of some other
varieties, e.g., Schubert varieties.
This talk is based on the joint work with T. Maeno, S. Murai and Y.
Numata. (TCPL 202) |

11:30 - 13:30 | Lunch (Vistas Dining Room) |

14:00 - 15:00 |
Peter Crooks: Integrable systems on families of Hessenberg varieties ↓ Kostant's invariant-theoretic version of the Toda lattice
has given rise to a powerful synergy of ideas from symplectic
geometry, algebraic geometry, and representation theory. An example
is the appearance of this Kostant-Toda lattice in calculations related
to the quantum cohomology of the flag variety. It is in this setting
that one compactifies the leaves of the Kostant-Toda lattice, thereby
obtaining a certain class of Hessenberg varieties. One might then
expect that the Kostant-Toda lattice can be defined on (the total
space of) a family of Hessenberg varieties. I will show this to be the
case, emphasizing the roles played by Slodowy slices and
Mishchenko-Fomenko theory. This represents joint work with Hiraku Abe. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:30 - 16:30 |
Ting Xue: Springer correspondence and Hessenberg varieties ↓ The Springer theory for reductive algebraic groups plays an important role in representation theory. It relates nilpotent orbits in the Lie algebra to irreducible representations of the Weyl group. We develop a Springer theory in the case of symmetric spaces using Fourier transform, which relates nilpotent orbits in this setting to irreducible representations of Hecke algebras with parameter -1. As an application we explain a general strategy for computing the cohomology of Hessenberg varieties. Examples of such varieties include classical objects in algebraic geometry: Jacobians, moduli spaces of vector bundles on curves with extra structure, and Fano varieties of linear subspaces in the intersection of two quadrics, etc. The talk is based on joint work with Tsao-Hsien Chen and Kari Vilonen. (TCPL 201) |

17:30 - 19:30 | Dinner (Vistas Dining Room) |

Friday, October 26 | |
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07:00 - 09:00 | Breakfast (Vistas Dining Room) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

11:30 - 12:00 |
Checkout by Noon ↓ 5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon. (Front Desk - Professional Development Centre) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |