TY - JOUR

T1 - The length and depth of algebraic groups

AU - Burness, Timothy C.

AU - Liebeck, Martin W.

AU - Shalev, Aner

PY - 2019/2

Y1 - 2019/2

N2 - Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups G = G0 > G1 > · · · > Gt = 1, where each Gi is a maximal connected subgroup of Gi−1. We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical Ru(G) and the dimension of a Borel subgroup B of the reductive quotient G/Ru(G). In particular, a simple algebraic group of rank r has length dim B +r, which gives a natural extension of a theorem of Solomon and Turull on
finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds 1/2 dim G.
We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G/R(G) is bounded above in terms
of the chain difference of G.

AB - Let G be a connected algebraic group. An unrefinable chain of G is a chain of subgroups G = G0 > G1 > · · · > Gt = 1, where each Gi is a maximal connected subgroup of Gi−1. We introduce the notion of the length (respectively, depth) of G, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group G in terms of the dimension of its unipotent radical Ru(G) and the dimension of a Borel subgroup B of the reductive quotient G/Ru(G). In particular, a simple algebraic group of rank r has length dim B +r, which gives a natural extension of a theorem of Solomon and Turull on
finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group G exceeds 1/2 dim G.
We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most 6 (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group G with soluble radical R(G), the dimension of G/R(G) is bounded above in terms
of the chain difference of G.

UR - http://www.scopus.com/inward/record.url?scp=85048041152&partnerID=8YFLogxK

U2 - 10.1007/s00209-018-2101-6

DO - 10.1007/s00209-018-2101-6

M3 - Article (Academic Journal)

AN - SCOPUS:85048041152

SN - 0025-5874

VL - 291

SP - 741

EP - 760

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

IS - 1-2

ER -