The Largest Coefficient of the Highest Root and the Second Smallest Exponent
- PDF / 313,171 Bytes
- 11 Pages / 439.37 x 666.142 pts Page_size
- 21 Downloads / 155 Views
(0123456789().,-volV)(0123456789().,-volV)
ORIGINAL PAPER
The Largest Coefficient of the Highest Root and the Second Smallest Exponent Tan Nhat Tran1 Received: 16 October 2019 / Revised: 8 August 2020 / Accepted: 9 September 2020 Ó Springer Japan KK, part of Springer Nature 2020
Abstract There are many different ways that the exponents of Weyl groups of irreducible root systems have been defined and put into practice. One of the most classical and algebraic definitions of the exponents is related to the eigenvalues of Coxeter elements. While the coefficients of the highest root when expressed as a linear combination of simple roots are combinatorial objects in nature, there are several results asserting relations between these exponents and coefficients. This study was conducted to give a uniform and bijective proof of the fact that the second smallest exponent of the Weyl group is one or two plus the largest coefficient of the highest root of the root system depending upon a simple condition on the root lengths. As a consequence, we obtain a necessary and sufficient condition for a root system to be of type G2 in terms of these numbers. Keywords Root system Highest root Weyl group Exponent
Mathematics Subject Classification Primary 17B22 Secondary 05A19
1 Introduction Assume that V ¼ R‘ with the standard inner product ð; Þ. For a 2 V, b 2 Vnf0g, denote ha; bi :¼ 2ða;bÞ ðb;bÞ . Let us denote by U an irreducible crystallographic root system in V. Let Uþ be a set of positive roots. With the notation D ¼ fa1 ; . . .; a‘ g, we have the set of simple roots of U with respect to Uþ . For any a; b 2 U, the P number ha; bi takes values in f0; 1; 2; 3g. For a ¼ ‘i¼1 di ai 2 Uþ , the height P of a is defined by htðaÞ :¼ ‘i¼1 di . Define the partial order on Uþ such that & Tan Nhat Tran [email protected] 1
Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
123
Graphs and Combinatorics
P P b a if a b 2 ‘i¼1 Z 0 ai for a; b 2 Uþ . Let h :¼ ‘i¼1 ci ai be the highest root of U with respect to the partial order, and we call ci ’s the coefficients of h. Denote by cmax :¼ maxfci j 1 i ‘g the largest coefficient. Let W be the Weyl group of U and let m1 ; m2 ; . . .; m‘ with m1 m2 m‘ be the exponents of W. The exponents of the Weyl group may have been originally defined in terms of the eigenvalues of Coxeter elements [7]. In addition, they can be defined as the degrees of the basic polynomial invariants of the Weyl group [6]. The multiset of the exponents has led to many important results and applications in study of Weyl arrangements, which are important examples of free arrangements, e.g., [14–16]. All of these above-mentioned definitions and applications are purely algebraic. Shapiro (unpublished), Steinberg [17], Kostant [10], Macdonald [12] and most recently also Abe–Barakat–Cuntz–Hoge–Terao [1] have shown that there is another way to derive the exponents, namely by the dual partition of the height distribution of Uþ . This approach not only gives
Data Loading...
