Multiarrangements of type B_2 and related topics
Dates：26 March 2024 -- 27 March 2024
Place：Room E301, 3rd floor, Building E, Faculty of Science, Toyonaka Campus, Osaka University (in person only)
Speakers：
Takuro Abe (Rikkyo University）
Hiraku Kawanoue (Chubu University)
Shuhei Tsujie (Hokkaido University of Education)
Yasuhide Numata (Hokkaido University)
Shota Maehara (Kyushu University)
Zixuan Wang (Hokkaido University)
Masahiko Yoshinaga (Osaka University）
Program:
26 March:
10:45--11:45 Masahiko Yoshinaga (Osaka University）
Title：Introduction to Catalan arrangement
13:30--14:30 Zixuan Wang (Hokkaido University）
Title：The Integral Expressions and Their Deformations for Derivations of Arrangements
14:45--15:45 Hiraku Kawanoue (Chubu University)
Title：A basis for the logarithmic vector field of the extended Catalan arrangement of type $B_2$
16:00--17:00 Shota Maehara (Kyushu University)
Title：On the existence of lower derivations for some multiarrangements of type B_2
27 March:
9:30--10:30 Shuhei Tsujie (Hokkaido University of Education)
Title： Freeness of Fuss-Catalan arrangements
10:45--11:45 Yasuhide Numata (Hokkaido University)
Title：On explicit description of bases of some line arrangements
13:30--14:30 Takuro Abe (Rikkyo University）
Title：Multi-Euler derivation
14:30--18:00 Free discussion
Title and abstract:
Takuro Abe (Rikkyo University）
Title:Multi-Euler derivation
Abstract:By using the affine connection, the Euler derivation gives an isomorphism between D(A,m) to itself, which is trivial. Wakamiko generalized this to the multi-Coxeter arrangement, and the role of the Euler derivation is played by so called the universal derivation. By using invariant theory and K. Saito's primitive derivation theory, this study has been promoted by Yoshinaga, Terao, Wakamiko, Mano, Roehrle, Stump and myself for reflectin arrangements. In this talk, we generalize this universal derivation to an arbitrary multi-arrangements that may not be related to the reflection arrangements, and introduce the multi-Euler derivation. In particular, we exhibit examples when the arrangement is of the type B_2 and the deleted A_3. This is a wok in progress with G. Roehrle and S. Wiesner.
Hiraku Kawanoue (Chubu University)
Title:A basis for the logarithmic vector field of the extended Catalan arrangement of type $B_2$
Abstract:In arXiv: 2309.01287, Feigin, Wang and Yoshinaga constructed explicitly a candidate of a basis
for the logarithmic vector field of the extended Catalan arrangement of type $B_2$.
I will talk about their construction and show that it gives a basis.
Shota Maehara (Kyushu University)
Title:On the existence of lower derivations for some multiarrangements of type B_2
Abstract:In this talk, let us consider only the Coxeter multiarrangements of type B_2 in a 2-dimensional vector space over a field of characteristic zero whose defining polynomials are xy(x-y)(x+y).
At first, for some multiplicities m=(m_1,m_2,m_3,m_3+n) satisfying n < 3, we show an explicit description of a basis for the derivation module D(m). And then by using these bases, we show necessary and sufficient conditions where exists a lower derivation for the basis of D(m), or the difference of the exponents is greater than 1, for some special multiplicities with n < 4.
The essential part in this talk is a joint work with Yasuhide Numata.
Yasuhide Numata (Hokkaido University)
Title:On explicit description of bases of some line arrangements.
Abstract:It is know that all modules of logarithmic derivations with respect to multi line arrangements
are free modules of rank two. Hence to give a basis for the module is one of important topics. In this talk,
we consider a basis for the module for multi Coxeter arrangements of type $B_2$, and discuss description of it.
This is based on the Joint work with Shota Maehara.
Shuhei Tsujie (Hokkaido University of Education)
Title: Freeness of Fuss-Catalan arrangements
Abstract: It is well known that the numbers of the chambers of the Catalan arrangements are given by the Catalan numbers.
The Fuss-Catalan numbers $A_{\ell}(m,r)$ are a two-parameter generalization of the Catalan numbers. When $r=1$ they give
the numbers of chambers of the extended Catalan arrangements. Recently, Deshpande, Menon, and Sarkar introduced a family
of arrangements such that the numbers of chambers of them are given by the Fuss-Catalan numbers with $r=2$. We call these
arrangements the Fuss-Catalan arrangements. There are some relations between the extended Catalan arrangements and the
Fuss-Catalan arrangements. For example, if we evaluate the characteristic polynomial of the Fuss-Catalan arrangement at
$t+1$, then we obtain the characteristic polynomial of the corresponding extended Catalan arrangement, and hence the
characteristic polynomial of the Fuss-Catalan arrangement decomposes into the product of linear polynomials over the
integer ring. This kind of phenomenon can be explained by using gain graphs. In terms of gain graphs, we will show that
freeness of the extended Catalan arrangements implies freeness of the Fuss-Catalan arrangements. Moreover, we will
introduce a recent attempt to give a basis for the Fuss-Catalan arrangements by using $q$-integral. This study is based
on joint work with Daisuke Suyama and Michele Torielli.
Zixuan Wang (Hokkaido University)
Title: The Integral Expressions and Their Deformations for Derivations of Arrangements
Abstract:The basis of Coxeter arrangements of type A can be constructed by Bandlow-Musiker's integral formula,
as detailed in the works of Bandlow, Musiker, Suyama, and Yoshinaga.
Abe-Suyama constructed the basis for the extended Catalan and Shi arrangements of type A_2.
Suyama-Yoshinaga used the discrete analogue of integral expressions to construct the explicit basis for the Catalan and Shi arrangements of type A.
In this talk, we will construct the basis for some multiarrangements by using integral expressions.
Specifically, we will explore the application of deformations of integral expressions to the Catalan arrangements of type B_2.
This talk is based on the joint work with Misha Feigin and Masahiko Yoshinaga.
Masahiko Yoshinaga (Osaka University）
Title:Introduction to Catalan arrangement
Abstract:TBA
世話人：
Takuro Abe (Rikkyo University)
Masahiko Yoshinaga (Osaka University)
Supported by:
JSPS KAKENHI Grant Number 23H00081
JSPS KAKENHI Grant Number 23K17298