Mask Formulas for Cograssmannian Kazhdan-Lusztig Polynomials

Jones, Brant; Woo, Alexander

doi:10.1007/s00026-012-0172-3

Cited by 7 publications

(12 citation statements)

References 49 publications

(61 reference statements)

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“…We will show next that the resolution

{\hat{S}}^{w}

is isomorphic to a particular Bott–Samelson resolution of

S^{w}

. We use Magyar's construction of Bott–Samelson resolutions as presented in Section 5 of . For every presentation

w = w_{1} \dots w_{l (w)}

as a reduced word, i.e.…”

Section: The Resolution Of the Flag Schubert Varietiesmentioning

confidence: 99%

“…Denote by BBS w the (bubblesort) Bott–Samelson resolution induced by this presentation of w . The projection map between BBS w and

{normalFl}_{1, 2, \dots, n} false(E false)

that gives the resolution of

S^{w}

is defined by (compare ):

[V_{1}, \dots, V_{l (w)}] \to (V_{p (1)}, \dots, V_{p (i)}, \dots, E)

where

p (i)

for

i \leq n - 1

is the total number of transpositions needed to bring

w (n)

to the last spot, …,

w (i + 1)

to spot

i + 1

. In other words:

p false(i false) = \sum_{j = i + 1}^{n} # false{1 \leq k \leq n false| w (k) > w (j) false}

and this is also the index of last occurence of

s_{i}

in the bubblesort reduced word for w .…”

Section: The Resolution Of the Flag Schubert Varietiesmentioning

confidence: 99%

“…Denote by BBS the (bubblesort) Bott-Samelson resolution induced by this presentation of . The projection map between BBS and Fl 1,2,…, ( ) that gives the resolution of is defined by (compare [9]):…”

Section: The Resolution Of the Flag Schubert Varietiesmentioning

confidence: 99%

“…It is then less than a surprise that the bioriented flag resolution is isomorphic to the Bott-Samelson resolution corresponding to the "bubblesort" reduced word decomposition of the permutation that defines the Schubert variety. Following the anonymous referee's suggestion, we prove this using the description of Bott-Samelson resolutions from [9].…”

Section: Introductionmentioning

confidence: 98%

“…In , Magyar gave a different construction of the Bott–Samelson resolutions, realizing them as subspaces of a product of Grassmannians. A particularly explicit description of Magyar's construction for

G L_{n} false(double-struckC false)

using incidence relations was given by Jones and Woo in . One common feature of all Bott–Samelson resolutions of a fixed Schubert variety determined by a permutation w is that they use a total of

l (w)

Grassmannian spaces.…”

Section: Introductionmentioning

confidence: 99%

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Bioriented flags and resolutions of Schubert varieties

Cibotaru

2020

Mathematische Nachrichten

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We use incidence relations running in two directions in order to construct a Kempf–Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott–Samelson resolutions. The second process led to the introduction of W‐flag varieties, algebro‐geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.

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“…We will show next that the resolution

{\hat{S}}^{w}

is isomorphic to a particular Bott–Samelson resolution of

S^{w}

. We use Magyar's construction of Bott–Samelson resolutions as presented in Section 5 of . For every presentation

w = w_{1} \dots w_{l (w)}

as a reduced word, i.e.…”

Section: The Resolution Of the Flag Schubert Varietiesmentioning

confidence: 99%

“…Denote by BBS w the (bubblesort) Bott–Samelson resolution induced by this presentation of w . The projection map between BBS w and

{normalFl}_{1, 2, \dots, n} false(E false)

that gives the resolution of

S^{w}

is defined by (compare ):

[V_{1}, \dots, V_{l (w)}] \to (V_{p (1)}, \dots, V_{p (i)}, \dots, E)

where

p (i)

for

i \leq n - 1

is the total number of transpositions needed to bring

w (n)

to the last spot, …,

w (i + 1)

to spot

i + 1

. In other words:

p false(i false) = \sum_{j = i + 1}^{n} # false{1 \leq k \leq n false| w (k) > w (j) false}

and this is also the index of last occurence of

s_{i}

in the bubblesort reduced word for w .…”

Section: The Resolution Of the Flag Schubert Varietiesmentioning

confidence: 99%

Section: The Resolution Of the Flag Schubert Varietiesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

G L_{n} false(double-struckC false)

using incidence relations was given by Jones and Woo in . One common feature of all Bott–Samelson resolutions of a fixed Schubert variety determined by a permutation w is that they use a total of

l (w)

Grassmannian spaces.…”

Section: Introductionmentioning

confidence: 99%

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Bioriented flags and resolutions of Schubert varieties

Cibotaru

2020

Mathematische Nachrichten

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Conjectures About Certain Parabolic Kazhdan–Lusztig Polynomials

Lapid¹

2018

Simons Symposia

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Rhombic tilings and Bott–Samelson varieties

Escobar

Pechenik

Tenner

et al. 2017

Proc. Amer. Math. Soc.

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S. Elnitsky (1997) gave an elegant bijection between rhombic tilings of 2n-gons and commutation classes of reduced words in the symmetric group on n letters. P. Magyar (1998) found an important construction of the Bott-Samelson varieties introduced by H.C. Hansen (1973) and M. Demazure (1974). We explain a natural connection between S. Elnitsky's and P. Magyar's results. This suggests using tilings to encapsulate Bott-Samelson data (in type A). It also indicates a geometric perspective on S. Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by B. Tenner (2006).

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Mask Formulas for Cograssmannian Kazhdan-Lusztig Polynomials

Cited by 7 publications

References 49 publications

Bioriented flags and resolutions of Schubert varieties

Bioriented flags and resolutions of Schubert varieties

Conjectures About Certain Parabolic Kazhdan–Lusztig Polynomials

Rhombic tilings and Bott–Samelson varieties

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