Rank-generating functions and Poincaré polynomials of lattices in finite orthogonal space of even characteristic

Abstract

The geometry of classical groups over finite fields is widely used in many fields. In this paper, we study the rank-generating function, the characteristic polynomial, and the Poincaré polynomial of lattices generated by the orbits of subspaces under finite orthogonal groups of even characteristic. We also determine their expressions.

Key words： Lattices; orthogonal space of even characteristic; rank-generating function; characteristic polynomial; Poincaré polynomial

References

1 AignerM. Combinatorial Theory. Grundlehren der Mathematischen Wissenschaften, Vol 234. Berlin: Springer-Verlag, 1979

2 Gao Y, You H. Lattices generated by orbits of subspaces under finite singular classical groups and its characteristic polynomials. Comm Algebra 2003; 31(6): 2927–2950

3 Huo Y J, Liu Y S, Wan Z X. Lattices generated by transitive sets of subspaces under finite classical groups, II. The orthogonal case of odd characteristic. Comm.Algebra 1992; 20(9): 2685–2727

4 Huo Y J, Wan Z X. On the geometricity of lattices generated by orbits of subspaces under finite classical groups. J Algebra 2001; 243(1): 339–359

5 Huo L F, Zhao Y B, Huo Y J. Critical problems of nonsingular subspaces in finite orthogonal spaces of even characteristic. Adv Math (China) 2014; 43(6): 824–834

6 Li Y, Zhao Y B, Huo Y J. Rank-generating functions and characteristic polynomials of lattice generated by orbits of subspaces in finite orthogonal space. Appl Math J Chinese Univ Ser A 2022; 37(3): 337–344

7 StanleyR P. Enumerative Combinatorics, Vol 1, 2nd ed. Cambridge: Cambridge University Press, 2012

8 WanZ X. Geometry of Classical Groups over Finite Fields, 2nd ed. Beijing: Science Press, 2002

9 WanZ XHuoY J. Lattices Generated by Orbits of Subspaces under Finite Classical Groups, 2nd Ed. Beijing: Science Press, 2004 (in Chinese)