抄録
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.
| 元の言語 | English |
|---|---|
| ページ（範囲） | 201-227 |
| ページ数 | 27 |
| ジャーナル | Journal of the London Mathematical Society |
| 巻 | 96 |
| 発行部数 | 1 |
| DOI | |
| 出版物ステータス | Published - 2017 8 1 |
| 外部発表 | Yes |
ASJC Scopus subject areas
- Mathematics(all)
これを引用
A comparison of Newton-Okounkov polytopes of Schubert varieties. / Fujita, Naoki; Oya, Hironori.
：: Journal of the London Mathematical Society, 巻 96, 番号 1, 01.08.2017, p. 201-227.研究成果: Article
}
TY - JOUR
T1 - A comparison of Newton-Okounkov polytopes of Schubert varieties
AU - Fujita, Naoki
AU - Oya, Hironori
PY - 2017/8/1
Y1 - 2017/8/1
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.
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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 -