Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

Fujita, Naoki; Naito, Seiji

doi:10.1007/s00209-016-1709-7

Cited by 26 publications

(58 citation statements)

References 20 publications

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“…Remark that

{{normalΞ}_{λ, w}^{up} false(b false) ∣ b \in {scriptB}_{w} false(λ false)}

is a

double-struckC

‐basis of

H^{0} false(X (w), L_{λ} false)

, and that

- {\overset{v}{true}}_{i}^{high} false(Ξ_{λ, w}^{up} (π_{λ} false(b false)) / τ_{λ} false) = {normalΨ}_{boldi} false(b false)

b \in {\overset{B}{true}}_{w} false(λ false)

, are all distinct. Hence we see by Proposition that

{v_{boldi}^{low} (σ / τ_{λ}) ∣ σ \in H^{0} (X false(w false), {scriptL}_{λ}) ∖ {0}} = {- {\tilde{v}}_{boldi}^{high} {false(σ / τ_{λ} false)}^{op} ∣ σ \in H^{0} (X false(w false), {scriptL}_{λ}) ∖ {0}} .

This implies the assertion by the definition of Newton–Okounkov bodies (see also the proof of [, Corollary 4.2]).

□

…”

Section: Resultssupporting

confidence: 61%

“…Proof Part (1) is an immediate consequence of the definition of

B^{up} false(λ false)

and . Parts (2), (4) are proved in a way similar to the proof of [, Lemma 4.5]. Since

U^{-}

is regarded as an open subvariety of

G / B

, we have

false(τ / τ_{λ} false) |_{U^{-}} \neq 0

for all nonzero sections

τ \in H^{0} false(G / B, L_{λ} false)

, which implies part (3).…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 69%

“…By condition

{(iii)}_{l}^{'}

and Proposition , we obtain the following (see, for instance, the proof of [, Proposition 2.8]). Proposition For all

i \in I

b \in B (\infty)

, and

k \in {double-struckZ}_{⩾ 0}

{normalΞ}^{low} false(b false) \cdot f_{i}^{k} \in {double-struckC}^{\times} {normalΞ}^{low} false({false({\tilde{f}}_{i}^{*} false)}^{k} b false) + \sum_{\begin{matrix} b^{'} \in scriptB (\infty); wt (b^{'}) = wt ({({\overset{f}{true}}_{i}^{*})}^{k} b), \\ ε_{i}^{*} (b^{'}) > ε_{i}^{*} ({({\overset{f}{true}}_{i}^{*})}^{k} b) \end{matrix}} C {normalΞ}^{low} false(b^{'} false) .

…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 88%

“…Remark By [, Theorem 3.10; , Section 1], we obtain a system of explicit linear inequalities defining the string polytope

Δ_{boldi}^{false(λ, w false)}

. In addition, under a certain positivity assumption on

boldi

, a system of explicit linear inequalities defining the Nakashima–Zelevinsky polyhedral realization

{\tilde{normalΔ}}_{boldi}^{false(λ, w false)}

is given by [, Theorem 4.1; , Proposition 3.1] (see also [, Corollary 5.3]).…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 99%

“…In this section, we first review the definitions and properties of perfect bases of the space [9] and [21]. Let P ⊂ t * be the weight lattice of g, and P + ⊂ P the set of dominant integral weights.…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 99%

See 4 more Smart Citations

A comparison of Newton–Okounkov polytopes of Schubert varieties

Fujita

Oya

2017

Journal of London Math Soc

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A Newton-Okounkov body is a convex body constructed from a polarized variety with a valuation on its function field. Kaveh (resp., 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 (resp., 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 (resp., 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.Comment: 21 pages, to appear in J. London Math. Soc. (2

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“…Remark that

{{normalΞ}_{λ, w}^{up} false(b false) ∣ b \in {scriptB}_{w} false(λ false)}

is a

double-struckC

‐basis of

H^{0} false(X (w), L_{λ} false)

, and that

- {\overset{v}{true}}_{i}^{high} false(Ξ_{λ, w}^{up} (π_{λ} false(b false)) / τ_{λ} false) = {normalΨ}_{boldi} false(b false)

b \in {\overset{B}{true}}_{w} false(λ false)

, are all distinct. Hence we see by Proposition that

{v_{boldi}^{low} (σ / τ_{λ}) ∣ σ \in H^{0} (X false(w false), {scriptL}_{λ}) ∖ {0}} = {- {\tilde{v}}_{boldi}^{high} {false(σ / τ_{λ} false)}^{op} ∣ σ \in H^{0} (X false(w false), {scriptL}_{λ}) ∖ {0}} .

This implies the assertion by the definition of Newton–Okounkov bodies (see also the proof of [, Corollary 4.2]).

□

…”

Section: Resultssupporting

confidence: 61%

“…Proof Part (1) is an immediate consequence of the definition of

B^{up} false(λ false)

and . Parts (2), (4) are proved in a way similar to the proof of [, Lemma 4.5]. Since

U^{-}

is regarded as an open subvariety of

G / B

, we have

false(τ / τ_{λ} false) |_{U^{-}} \neq 0

for all nonzero sections

τ \in H^{0} false(G / B, L_{λ} false)

, which implies part (3).…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 69%

“…By condition

{(iii)}_{l}^{'}

and Proposition , we obtain the following (see, for instance, the proof of [, Proposition 2.8]). Proposition For all

i \in I

b \in B (\infty)

, and

k \in {double-struckZ}_{⩾ 0}

{normalΞ}^{low} false(b false) \cdot f_{i}^{k} \in {double-struckC}^{\times} {normalΞ}^{low} false({false({\tilde{f}}_{i}^{*} false)}^{k} b false) + \sum_{\begin{matrix} b^{'} \in scriptB (\infty); wt (b^{'}) = wt ({({\overset{f}{true}}_{i}^{*})}^{k} b), \\ ε_{i}^{*} (b^{'}) > ε_{i}^{*} ({({\overset{f}{true}}_{i}^{*})}^{k} b) \end{matrix}} C {normalΞ}^{low} false(b^{'} false) .

…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 88%

“…Remark By [, Theorem 3.10; , Section 1], we obtain a system of explicit linear inequalities defining the string polytope

Δ_{boldi}^{false(λ, w false)}

. In addition, under a certain positivity assumption on

boldi

, a system of explicit linear inequalities defining the Nakashima–Zelevinsky polyhedral realization

{\tilde{normalΔ}}_{boldi}^{false(λ, w false)}

is given by [, Theorem 4.1; , Proposition 3.1] (see also [, Corollary 5.3]).…”

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 99%

Section: Kashiwara Crystal Bases and Perfect Basesmentioning

confidence: 99%

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A comparison of Newton–Okounkov polytopes of Schubert varieties

Fujita

Oya

2017

Journal of London Math Soc

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Polyhedral realizations of crystal bases and convex-geometric Demazure operators

Fujita

2019

Sel. Math. New Ser.

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Combinatorics of Canonical Bases Revisited: String Data in Type A

Genz

Koshevoy

Schumann

2021

Transformation Groups

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We give a formula for the crystal structure on the integer points of the string polytopes and the *-crystal structure on the integer points of the string cones of type A for arbitrary reduced words. As a byproduct, we obtain defining inequalities for Nakashima–Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara *-involution on string data for a special choice of reduced word.

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Newton–Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases

Cited by 26 publications

References 20 publications

A comparison of Newton–Okounkov polytopes of Schubert varieties

A comparison of Newton–Okounkov polytopes of Schubert varieties

Polyhedral realizations of crystal bases and convex-geometric Demazure operators

Combinatorics of Canonical Bases Revisited: String Data in Type A

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