TY - JOUR

T1 - A comparison of Newton-Okounkov polytopes of Schubert varieties

AU - Fujita, Naoki

AU - Oya, Hironori

N1 - Funding Information:
The work of the first author was supported by Grant-in-Aid for JSPS Fellows (No. 16J00420). The work of the second author was supported by Grant-in-Aid for JSPS Fellows (No. 15J09231) and the Program for Leading Graduate Schools, MEXT, Japan.

PY - 2017/8

Y1 - 2017/8

N2 - A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (respectively, the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (respectively, the Nakashima-Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott-Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (respectively, by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation.

AB - A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (respectively, the first author and Naito) proved that the Newton-Okounkov body of a Schubert variety associated with a specific valuation is identical to the Littelmann string polytope (respectively, the Nakashima-Zelevinsky polyhedral realization) of a Demazure crystal. These specific valuations are defined algebraically to be the highest term valuations with respect to certain local coordinate systems on a Bott-Samelson variety. Another class of valuations, which is geometrically natural, arises from some sequence of subvarieties of a polarized variety. In this paper, we show that the highest term valuation used by Kaveh (respectively, by the first author and Naito) and the valuation coming from a sequence of specific subvarieties of the Schubert variety are identical on a perfect basis with some positivity properties. The existence of such a perfect basis follows from a categorification of the negative part of the quantized enveloping algebra. As a corollary, we prove that the associated Newton-Okounkov bodies coincide through an explicit affine transformation.

UR - http://www.scopus.com/inward/record.url?scp=85021419245&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021419245&partnerID=8YFLogxK

U2 - 10.1112/jlms.12059

DO - 10.1112/jlms.12059

M3 - Article

AN - SCOPUS:85021419245

VL - 96

SP - 201

EP - 227

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

SN - 0024-6107

IS - 1

ER -