TY - JOUR

T1 - On A12 restrictions of Weyl arrangements

AU - Abe, Takuro

AU - Terao, Hiroaki

AU - Tran, Tan Nhat

N1 - Funding Information:
The current paper is an improvement of the third author’s Master’s thesis, written under the supervision of the second author at Hokkaido University in 2017. At that time, the first main result (Theorem ) was only proved by a case-by-case method. The second author was supported by JSPS Grants-in-Aid for basic research (A) No. 24244001. The third author was partially supported by the scholarship program of Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT) No. 142506 and is currently supported by JSPS Research Fellowship for Young Scientists No. 19J12024.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020

Y1 - 2020

N2 - Let A be a Weyl arrangement in an ℓ-dimensional Euclidean space. The freeness of restrictions of A was first settled by a case-by-case method by Orlik and Terao (Tôhoku Math J 52: 369–383, 1993), and later by a uniform argument by Douglass (Represent Theory 3: 444–456, 1999). Prior to this, Orlik and Solomon (Proc Symp Pure Math Amer Math Soc 40(2): 269–292, 1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik et al. (Adv Stud Pure Math 8: 461–77, 1986) asserts that the exponents of any A1 restriction, i.e., the restriction of A to a hyperplane, are given by { m1, … , mℓ-1} , where exp (A) = { m1, … , mℓ} with m1≤ ⋯ ≤ mℓ. As a next step towards conceptual understanding of the restriction exponents, we will investigate the A12 restrictions, i.e., the restrictions of A to the subspaces of type A12. In this paper, we give a combinatorial description of the exponents and describe bases for the modules of derivations of the A12 restrictions in terms of the classical notion of related roots by Kostant (Proc Nat Acad Sci USA 41:967–970, 1955).

AB - Let A be a Weyl arrangement in an ℓ-dimensional Euclidean space. The freeness of restrictions of A was first settled by a case-by-case method by Orlik and Terao (Tôhoku Math J 52: 369–383, 1993), and later by a uniform argument by Douglass (Represent Theory 3: 444–456, 1999). Prior to this, Orlik and Solomon (Proc Symp Pure Math Amer Math Soc 40(2): 269–292, 1983) had completely determined the exponents of these arrangements by exhaustion. A classical result due to Orlik et al. (Adv Stud Pure Math 8: 461–77, 1986) asserts that the exponents of any A1 restriction, i.e., the restriction of A to a hyperplane, are given by { m1, … , mℓ-1} , where exp (A) = { m1, … , mℓ} with m1≤ ⋯ ≤ mℓ. As a next step towards conceptual understanding of the restriction exponents, we will investigate the A12 restrictions, i.e., the restrictions of A to the subspaces of type A12. In this paper, we give a combinatorial description of the exponents and describe bases for the modules of derivations of the A12 restrictions in terms of the classical notion of related roots by Kostant (Proc Nat Acad Sci USA 41:967–970, 1955).

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U2 - 10.1007/s10801-020-00979-8

DO - 10.1007/s10801-020-00979-8

M3 - Article

AN - SCOPUS:85091071093

JO - Journal of Algebraic Combinatorics

JF - Journal of Algebraic Combinatorics

SN - 0925-9899

ER -