We study the poset of Borel congruence classes of symmetric matrices ordered by containment of closures. We give a combinatorial description of this poset and calculate its rank function. We discuss the relation between this poset and the Bruhat poset of involutions of the symmetric group.
Bibliographical noteFunding Information:
The second author was also supported by the Swiss National Science Foundation , 2009.
It is a pleasure for us to thank Dr. Anna Melnikov, whose work  gave the starting point for this paper. We are grateful to Prof. Ron Adin for many helpful discussions. We would like to express a special gratitude to Prof. Lex Renner for providing us very useful information about the Bruhat poset and for answering our questions. Also, we are grateful to Prof. Yuval Roichman and to Prof. Uzi Vishne for helpful discussions. The second author would like to thank the Faculty of Mathematics, Technion, Israel, and the Department of Mathematics, University of Geneva, Switzerland, for the hospitality and for the financial support.
- Bruhat poset
- Involutions of the symmetric group
- Partial permutations