Three-Dimensional Mirror Self-Symmetry of the Cotangent Bundle of the Full Flag Variety

Rimányi, Richárd; Smirnov, Andrey; Varchenko, Alexander; Zhou, Zijun

doi:10.3842/sigma.2019.093

Cited by 35 publications

(39 citation statements)

References 42 publications

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“…This new property of weight functions plays an important role in a follow‐up paper [48] in connection with elliptic stable envelopes. It is worth pointing out that the normalization (11.1) also makes the R‐matrix property of (6.9) unified:

{\overset{w}{true}}_{s_{k} ω}^{'} = {\overset{w}{true}}_{ω}^{'} \cdot \frac{δ (\frac{μ_{ω^{- 1} (k + 1)}}{μ_{ω^{- 1} (k)}}, \frac{z_{k + 1}}{z_{k}})}{δ (\frac{μ_{ω^{- 1} (k)}}{μ_{ω^{- 1} (k + 1)}}, h)} + s_{k}^{z} {\overset{w}{true}}_{ω}^{'} \cdot \frac{δ (\frac{z_{k}}{z_{k + 1}}, h)}{δ (\frac{μ_{ω^{- 1} (k)}}{μ_{ω^{- 1} (k + 1)}}, h)} .

There is, however, an essential difference between (11.2) and (11.3): the latter holds for the weight functions themselves, while the former only holds for the cosets

[{\overset{w}{true}}^{'}]

{\hat{boldw}}^{'}

functions (that is, after the restriction).…”

Section: A Tale Of Two Recursions For Weight Functionsmentioning

confidence: 99%

“…A remarkable by‐product of the fact that the

E (X_{ω})

classes satisfy two seemingly unrelated recursions is the fact that weight functions, besides satisfying the known R‐matrix recursions, also satisfy a so far unknown recursion coming from the Bott–Samelson induction. This will be presented in Section 11, and will be interpreted as the R‐matrix relation for the 3D mirror dual variety in a follow up paper (for a special case see [48]).…”

Section: Introductionmentioning

confidence: 99%

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Elliptic classes of Schubert varieties via Bott–Samelson resolution

Rimányi

Weber

2020

Journal of Topology

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Based on recent advances on the relation between geometry and representation theory, we propose a new approach to elliptic Schubert calculus. We study the equivariant elliptic characteristic classes of Schubert varieties of the generalized full flag variety G/B. For this first we need to twist the notion of elliptic characteristic class of Borisov–Libgober by a line bundle, and thus allow the elliptic classes to depend on extra variables. Using the Bott–Samelson resolution of Schubert varieties we prove a BGG‐type recursion for the elliptic classes, and study the Hecke algebra of our elliptic BGG operators. For G=prefixGLnfalse(double-struckCfalse) we find representatives of the elliptic classes of Schubert varieties in natural presentations of the K theory ring of G/B, and identify them with the Tarasov–Varchenko weight function. As a byproduct we find another recursion, different from the known R‐matrix recursion for the fixed point restrictions of weight functions. On the other hand the R‐matrix recursion generalizes for arbitrary reductive group G.

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{\overset{w}{true}}_{s_{k} ω}^{'} = {\overset{w}{true}}_{ω}^{'} \cdot \frac{δ (\frac{μ_{ω^{- 1} (k + 1)}}{μ_{ω^{- 1} (k)}}, \frac{z_{k + 1}}{z_{k}})}{δ (\frac{μ_{ω^{- 1} (k)}}{μ_{ω^{- 1} (k + 1)}}, h)} + s_{k}^{z} {\overset{w}{true}}_{ω}^{'} \cdot \frac{δ (\frac{z_{k}}{z_{k + 1}}, h)}{δ (\frac{μ_{ω^{- 1} (k)}}{μ_{ω^{- 1} (k + 1)}}, h)} .

There is, however, an essential difference between (11.2) and (11.3): the latter holds for the weight functions themselves, while the former only holds for the cosets

[{\overset{w}{true}}^{'}]

{\hat{boldw}}^{'}

functions (that is, after the restriction).…”

Section: A Tale Of Two Recursions For Weight Functionsmentioning

confidence: 99%

“…A remarkable by‐product of the fact that the

E (X_{ω})

Section: Introductionmentioning

confidence: 99%

Elliptic classes of Schubert varieties via Bott–Samelson resolution

Rimányi

Weber

2020

Journal of Topology

Self Cite

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“…These identities are, indeed, true for the theta-function, ξ(z) = θ w (z). The identities are four-term and each term is a product of six θ-functions, they look similar to the ones appearing in description of the elliptic R-matrices in [44,45]. The first identities (3.4), (3.6) emerged even earlier in [46, eq.…”

Section: Jhep08(2020)150 3 Symmetric Polynomials From Els-functionsmentioning

confidence: 68%

Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function

Awata

Kanno

Миронов

et al. 2020

J. High Energ. Phys.

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As a development of [1], we note that the ordinary Shiraishi functions have an insufficient number of parameters to describe generic eigenfunctions of double elliptic system (Dell). The lacking parameter can be provided by substituting elliptic instead of the ordinary Gamma functions in the coefficients of the series. These new functions (ELS-functions) are conjectured to be functions governed by compactified DIM networks which can simultaneously play the three roles: solutions to non-stationary Dell equations, Dell conformal blocks with the degenerate field (surface operator) insertion, and the corresponding instanton sums in 6d SUSY gauge theories with adjoint matter. We describe the basics of the corresponding construction and make further conjectures about the various limits and dualities which need to be checked to make a precise statement about the Dell description by double-periodic network models with DIM symmetry. We also demonstrate that the ELS-functions provide symmetric polynomials, which are an elliptic generalization of Macdonald ones, and compute the generation function of the elliptic genera of the affine Laumon spaces. In the particular U(1) case, we find an explicit plethystic formula for the 6d partition function, which is a non-trivial elliptic generalization of the (q, t) Nekrasov-Okounkov formula from 5d.

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“…The examples of 3d-mirror symmetry for elliptic stable envelopes (which were not yet available at the time of the first release of this paper) can be found in [31,32]. In particular, the case of cotangent bundles over complete flag varieties of A n type [32] is another interesting example of 3d-selfdual symplectic variety.…”

Section: 4mentioning

confidence: 99%

“…The scheme X X is a (symmetric) power of E and every such section can be described through the theta functions as a certain symmetric combination of the elliptic Chern roots of the tautological bundles. The stable envelopes of the fixed points presented in the off-shell form are also known as weight functions in literature, see for example [32] and references there.…”

Section: 14mentioning

confidence: 99%