Let \(G/ B\) be a generalized flag variety where \(G\) is a complex, reductive algebraic group and \(B\) is a Borel subgroup.
The study of orbits of a symmetric subgroup on the flag variety was initiated in [G. Lusztig and D. A. Vogan jun., Invent. Math. 71, 365–379 (1983; Zbl 0544.14035)], where the singularities of closures of the orbits were related to characters of particular infinite-dimensional representations of a certain real form \(G_R\) of \(G\). Since then, there has been a stream of results on the combinatorics and geometry of these orbit closures.
Notably, R. W. Richardson and T. A. Springer [Geom. Dedicata 35, No. 1–3, 389–436 (1990; Zbl 0704.20039)] gave a description of the partial order given by inclusions of orbit closures, and M. Brion [Comment. Math. Helv. 76, No. 2, 263–299 (2001; Zbl 1043.14012)] studied general properties of their singularities, showing that, in many cases, including the one addressed in this paper, all these orbit closures are normal and Cohen-Macaulay with rational singularities.
In this paper, the authors initiate a combinatorial approach, backed by explicit commutative algebra computations, to the study of the singularities of these orbit closures for the symmetric subgroup \(K=\mathrm{GL}_\times\mathrm{GL}_q\) of block diagonal matrices in \(\mathrm{GL}_n\), where \(n= p+q\).
For this symmetric subgroup \(K\) , the finitely many \(K\)-orbits \(O\) and their closures \(Y\) can be parametrized by \((p,q)\)-clans (see [T. Matsuki and T. Oshima, Prog. Math. 82, 147–175 (1990; Zbl 0746.22011), Theorem 4.1] and [A. Yamamoto, Represent. Theory 1, 329–404 (1997; Zbl 0887.22017), Theorem 2.2.8]). These clans are partial matchings of vertices \({1,2,\dots, n}\), where each unmatched vertex is assigned a sign of + or -; the difference in the number of + and - signs must be \(p-q\).
The author were inspired by W. M. McGovern [J. Algebra 322, No. 8, 2709–2712 (2009; Zbl 1185.14040)] characterization of singular \(K\)-orbit closures in terms of pattern avoidance of clans. Roughly speaking, a clan \(\gamma\) contains a clan \(\theta\) if it is obtained by \(\theta\) by inserting \(+,-\) and couple of integers. For example \(134++2-431\) – contains \(1+2-21\). In the contraire case \(\gamma\) avoid \(\theta\). The main theorem of the cited McGovern work asserts that \(Y\) is smooth if and only if it avoids the patterns \(1+-1\); \(1-+1\); \(1212\); \(1+221\); \(1-221\); \(122+1\); \(122-1\); \(122331\).
On the other hand, in [A. Woo and B. J. Wyser, Int. Math. Res. Not. 2015, No. 24, 13148–13193 (2015; Zbl 1348.14120)] it is noted that the concept of avoidance is not sufficient to characterize Gorenstein property.
Suppose that \(P\) is any singularity mildness property, by which we mean a local property of varieties that holds on open subsets and is stable under smooth morphisms. Many singularity properties, such as being Gorenstein, being a local complete intersection (lci), being factorial, having Cohen-Macaulay rank \(\leq k\) , or having Hilbert-Samuel multiplicity \(\leq k\), satisfy these conditions. For such a \(P\) , consider two related problems:
(I) Which K-orbit closures \(Y\) are globally \(P\) ?
(II) What is the non-\(P\) -locus of \(Y\) ?
This paper gives a universal combinatorial language, interval pattern avoidance of clans, to answer these questions for any singularity mildness property, at the cost of potentially requiring an infinite number of patterns. This language is also useful for collecting and analyzing data and partial results. We present explicit equations for computing whether a property holds at a specific orbit \(O\) on an orbit closure \(Y\).
The non \(P\)-locus is closed, it is a union of \(K\)-orbit closures. Consequently, for any given clan \(\gamma\), (II) can be answered by finding a finite set of clans, namely those indexing the irreducible components of the non-\(P\)-locus. (I) asks if this set is nonempty.
This situation parallels that for Schubert varieties. In that setting, the first and third authors introduced interval pattern avoidance for permutations, showing that it provides a common perspective to study all reasonable singularity measures (see [A. Woo and A. Yong, Adv. Math. 207, No. 1, 205–220 (2006; Zbl 1112.14058); J. Algebra 320, No. 2, 495–520 (2008; Zbl 1152.14046)] and [A. Woo, Can. Math. Bull. 53, No. 4, 757–762 (2010; Zbl 1203.14055)]). This paper gives the first analogue of those results for \(K\)-orbit closures.
For most finer singularity mildness properties, answers to both (I) and (II) are unknown.