A new prehomogeneous vector space of characteristic p
This paper deals with a new type of prehomogeneous vector space of characteristic \(p\) which does not appear in the case of characteristic zero. The author has shown the following theorem: Let K be an algebraically closed field of characteristic 3. Let \(\Lambda_ 1, \Lambda_ 2, \Lambda_ 3\) denote all the fundamental dominant weights of GL(4). Then the K-dimension of the irreducible GL(4)-module V with the highest weight \(\Lambda_ 1+\Lambda_ 2\) is equal to 16, and it is denoted by V(16). In this paper, the following results are proved. (1) \((GL(4),\Lambda_ 1+\Lambda_ 2,V(16))\) is a regular irreducible prehomogeneous vector space. The degree of its irreducible relative invariant is 8, the associated character is \(\chi (g)=(\det (g))\) 6. (2) There exist only one 6-dimensional GL(4)-orbit and one 9-dimensional GL(4)-orbit in V(16). When \(m=7, 8\) or \(1\leq m\leq 5\), there are no m- dimensional GL(4)-orbits.
regular irreducible prehomogeneous vector space
scientific article; zbMATH DE number 4051792
scientific article