Details:
Title  Gr\"obner geometry of vertex decompositions and of flagged tableaux.  Author(s)  Allen Knutson, Ezra Miller, Alexander Yong  Type  Article in Journal  Abstract  We relate a classic algebrogeometric degeneration technique, dating at least to Hodge 1941 (J. London Math. Soc. 16: 245–255), to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a geometric vertex decomposition.
Our main example in this paper is the family of vexillary matrix Schubert varieties, whose ideals are also known as (onesided) ladder determinantal ideals. Using a diagonal term order to specify the (Gröbner) degeneration, we show that these have geometric vertex decompositions into simpler varieties of the same type. From this, together with the combinatorics of the pipe dreams of FominKirillov 1996 (Discr. Math. 153: 123–143), we derive a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, in terms of flagged setvalued tableaux. This unifies work of Wachs 1985 (J. Combin. Th. (A) 40: 276–289) on flagged tableaux, and Buch 2002 (Acta. Math. 189: 37–78) on setvalued tableaux, giving geometric meaning to both.
This work focuses on diagonal term orders, giving results complementary to those of KnutsonMiller 2005 (Ann. Math. 161: 1245–1318), where it was shown that the generating minors form a Gröbner basis for any antidiagonal term order and any matrix Schubert variety. We show here that under a diagonal term order, the only matrix Schubert varieties for which these minors form Gröbner bases are the vexillary ones, reaching an end toward which the ladder determinantal literature had been building.  ISSN  00754102; 14355345/e 
URL 
http://www.degruyter.com/view/j/crll.2009.2009.issue630/crelle.2009.033/crelle.2009.033.xml 
Language  English  Journal  J. Reine Angew. Math.  Volume  630  Pages  131  Publisher  De Gruyter, Berlin  Year  2009  Edition  0  Translation 
No  Refereed 
No 
