TY - JOUR

T1 - Gröbner geometry of vertex decompositions and of flagged tableaux

AU - Knutson, Allen

AU - Miller, Ezra

AU - Yong, Alexander

N1 - Funding Information:
We would like to thank David Eisenbud, Manoj Kummini, Vic Reiner, Bernd Sturm-fels and Alex Woo for helpful conversations. We thank Frank Sottile for directing us to the reference [Hod]. AK was supported by an NSF grant and EM was supported by NSF grants DMS-0304789 and DMS-0449102. This work was partially completed while AY was supported by NSF grant DMS-0601010 and while an NSERC supported visitor at the Fields Institute in Toronto, during the 2005 program ‘‘The Geometry of String Theory’’.

PY - 2009/5

Y1 - 2009/5

N2 - We relate a classic algebro-geometric 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 (one-sided) 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 Fomin-Kirillov 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 set-valued 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 set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of Knutson-Miller 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.

AB - We relate a classic algebro-geometric 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 (one-sided) 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 Fomin-Kirillov 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 set-valued 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 set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of Knutson-Miller 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.

UR - http://www.scopus.com/inward/record.url?scp=67650248593&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=67650248593&partnerID=8YFLogxK

U2 - 10.1515/CRELLE.2009.033

DO - 10.1515/CRELLE.2009.033

M3 - Article

AN - SCOPUS:67650248593

SP - 1

EP - 31

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 630

ER -