Combinatorial Models for the Variety of Complete Quadrics

We develop several combinatorial models that are useful in the study of the $SL_n$-variety $\mathcal{X}$ of complete quadrics. Barred permutations parameterize the fixed points of the action of a maximal torus $T$ of $SL_n$, while $\mu$-involutions parameterize the orbits of a Borel subgroup of $SL_n$. Using these combinatorial objects, we characterize the $T$-stable curves and surfaces on $\mathcal{X}$, compute the $T$-equivariant $K$-theory of $\mathcal{X}$, and describe a Bialynicki-Birula cell decomposition for $\mathcal{X}$. Furthermore, we give a computational characterization of the Bruhat order on Borel orbits in $\mathcal{X}$.

[1]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[2]  S. Banerjee,et al.  Equivariant $K$-theory of smooth projective spherical varieties , 2016, 1603.04926.

[3]  Eric Marberg,et al.  Involution words II: braid relations and atomic structures , 2016, 1601.02269.

[4]  Mahir Bilen Can,et al.  Wonderful symmetric varieties and Schubert polynomials , 2015, Ars Math. Contemp..

[5]  Mahir Bilen Can,et al.  Chains in weak order posets associated to involutions , 2014, J. Comb. Theory, Ser. A.

[6]  M. Can,et al.  Weak order on complete quadrics , 2012, 1201.2149.

[7]  A. Knutson.  A compactly supported formula for equivariant localization, and, simplicial complexes of Bialynicki-Birula decompositions , 2008, 0801.4092.

[8]  F. Incitti.  The Bruhat Order on the Involutions of the Symmetric Group , 2004 .

[9]  J. Carrell,et al.  Smooth points of T-stable varieties in G/B and the Peterson map , 2000, math/0005025.

[10]  T. A. Springer,et al.  Compactification of symmetric varieties , 1999 .

[11]  M. Brion.  The behaviour at infinity of the Bruhat decomposition , 1998 .

[12]  Michel Brion,et al.  Equivariant Chow groups for torus actions , 1997 .

[13]  D. A. Timashev.  Generalization Of The Bruhat Decomposition , 1995 .

[14]  T. A. Springer,et al.  Complements to ‘The Bruhat order on symmetric varieties’ , 1994 .

[15]  T. A. Springer,et al.  The Bruhat order on symmetric varieties , 1990 .

[16]  M. Goresky,et al.  On the geometry of quadrics and their degenerations , 1988 .

[17]  E. Strickland.  Schubert-type cells for complete quadrics , 1986 .

[18]  A. Białynicki-Birula.  Some theorems on actions of algebraic groups , 1973 .

[19]  G. Faltings.  Explicit resolution of local singularities of moduli-spaces. , 1997 .

[20]  A. Thorup,et al.  Complete bilinear forms , 1988 .

[21]  M. Brion,et al.  Sur la structure locale des variétés sphériques , 1987 .

[22]  Claudio Procesi,et al.  Complete symmetric varieties , 1983 .

[23]  I. Vainsencher.  Schubert Calculus for Complete Quadrics , 1982 .

[24]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .