Highest weights for truncated shifted Yangians and product monomial crystals

Kamnitzer, Joel; Tingley, Peter; Webster, Ben; Weekes, Alex; Yacobi, Oded

doi:10.4171/jca/32

Cited by 22 publications

(45 citation statements)

References 14 publications

(40 reference statements)

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“…Theorem ). In this paper, we prove our conjecture from . Theorem Let

boldR

be an integral set of parameters.…”

Section: Introductionmentioning

confidence: 75%

“…In , we formulated a conjectural answer using the product monomial crystal

B (R)

. This is a

frakturg

‐crystal whose elements are collections of rational monomials in variables

a_{i, k}

, where

i \in I

and

k \in Z

.…”

Section: Introductionmentioning

confidence: 99%

“…As explained in , this theorem is motivated from Nakajima's equivariant version of the Hikita conjecture. Thus, in proving this theorem, we have proved a weak form of the equivariant Hikita conjecture for the symplectic dual pair

{sans-serifGr}_{μ}^{λ}, M false(λ, μ false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Given a dominant weight

λ

and an integral set of parameters

boldR

of weight

λ

as above, following , we define the product monomial crystal

B (R)

B false(boldR false) = \prod_{i \in I, c \in R_{i}} B false(ϖ_{i}, c false)

In other words, for each parameter

c \in R_{i}

, we form its monomial crystal

B (ϖ_{i}, c)

and then take the product of all monomials appearing in all these crystals. In , we proved the following result as a consequence of the link between

B (R)

and graded quiver varieties. It also follows as an immediate corollary of the link between the monomial crystal and the parity KLRW algebra, see Theorem of the current paper.…”

Section: Introductionmentioning

confidence: 99%

“…Thus,

B (R)

is a crystal which depends on the set of parameters

boldR

and lies between the crystal of the irreducible representation and the crystal of the corresponding tensor product of fundamental representations (for an example see [, Section 2.6]).…”

Section: Introductionmentioning

confidence: 99%

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On category O for affine Grassmannian slices and categorified tensor products

Kamnitzer

Tingley

Webster

et al. 2019

Proc. London Math. Soc.

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Truncated shifted Yangians are a family of algebras which naturally quantize slices in the affine Grassmannian. These algebras depend on a choice of two weights λ and μ for a Lie algebra frakturg, which we will assume is simply laced. In this paper, we relate the category scriptO over truncated shifted Yangians to categorified tensor products: For a generic integral choice of parameters, category scriptO is equivalent to a weight space in the categorification of a tensor product of fundamental representations defined by the third author using KLRW algebras. We also give a precise description of category scriptO for arbitrary parameters using a new algebra which we call the parity KLRW algebra. In particular, we confirm the conjecture of the authors that the highest weights of category scriptO are in canonical bijection with a product monomial crystal depending on the choice of parameters. This work also has interesting applications to classical representation theory. In particular, it allows us to give a classification of simple Gelfand–Tsetlin modules of U(frakturgln) and its associated W‐algebras.

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“…Theorem ). In this paper, we prove our conjecture from . Theorem Let

boldR

be an integral set of parameters.…”

Section: Introductionmentioning

confidence: 75%

“…In , we formulated a conjectural answer using the product monomial crystal

B (R)

. This is a

frakturg

‐crystal whose elements are collections of rational monomials in variables

a_{i, k}

, where

i \in I

and

k \in Z

.…”

Section: Introductionmentioning

confidence: 99%

{sans-serifGr}_{μ}^{λ}, M false(λ, μ false)

.…”

Section: Introductionmentioning

confidence: 99%

“…Given a dominant weight

λ

and an integral set of parameters

boldR

of weight

λ

as above, following , we define the product monomial crystal

B (R)

B false(boldR false) = \prod_{i \in I, c \in R_{i}} B false(ϖ_{i}, c false)

In other words, for each parameter

c \in R_{i}

, we form its monomial crystal

B (ϖ_{i}, c)

and then take the product of all monomials appearing in all these crystals. In , we proved the following result as a consequence of the link between

B (R)

and graded quiver varieties. It also follows as an immediate corollary of the link between the monomial crystal and the parity KLRW algebra, see Theorem of the current paper.…”

Section: Introductionmentioning

confidence: 99%

“…Thus,

B (R)

is a crystal which depends on the set of parameters

boldR

and lies between the crystal of the irreducible representation and the crystal of the corresponding tensor product of fundamental representations (for an example see [, Section 2.6]).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On category O for affine Grassmannian slices and categorified tensor products

Kamnitzer

Tingley

Webster

et al. 2019

Proc. London Math. Soc.

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Coulomb branches of star-shaped quivers

Dimofte

Garner

2019

J. High Energ. Phys.

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We study the Coulomb branches of 3d N = 4 "star-shaped" quiver gauge theories and their deformation quantizations, by applying algebraic techniques that have been developed in the mathematics and physics literature over the last few years. The algebraic techniques supply an abelianization map, which embeds the Coulomb-branch chiral ring into a vastly simpler abelian algebra A. Relations among chiral-ring operators, and their deformation quantization, are canonically induced from the embedding into A. In the case of star-shaped quivers -whose Coulomb branches are related to Higgs branches of 4d N = 2 theories of Class S -this allows us to systematically verify known relations, to generalize them, and to quantize them. In the quantized setting, we find several new families of relations. arXiv:1808.05226v2 [hep-th] 7 Dec 2018In the initial work [12], the precise image of the embedding (1.4) was only identified in a handful of examples; however, at least in principle, a complete combinatorial construction of the image has since been described by Webster [18].In the case of T N,k theories, we will identify the putative generators of C[M C ] proposed by [21][22][23][24][25] (from a 4d Higgs-branch perspective) as elements of A. We will show how to explicitly verify and then quantize the conjectured relations among them.An important insight in the derivation of C[M 4d H ] chiral-ring relations in [24] was that various generators could be "diagonalized," as tensors for the SU (N ) k flavor symmetry. We find that the abelian algebra A plays a surprisingly important role in this diagonalization. In particular, the eigenvalues of the generators, which are complicated algebraic functions on the actual moduli space M C ≈ M 4d H , turn out to be extremely simple monomials in the algebra A. This allows the entire diagonalization procedure to be deformation-quantized.From the perspective of 4d Higgs branches, the fact that the chiral ring C[M 4d H ] admits a deformation quantization may not be obvious. However, this extra structure is completely natural (and physical) in 3d Coulomb branches. Indeed, in the recent mathematical/TQFT constructions of Coulomb branches [12,13,[15][16][17][18], one typically works with quantized algebras from the very beginning. In physical terms, the Poisson structure in the chiral ring of 3d N = 4 theories arises from topological descent in the Rozansky-Witten twist [36][37][38], and quantization comes from turning on an Omega background [39,40]. (See also [41,42]. An analogous quantization arising from an Omega background in four dimensions is familiar from [43][44][45][46][47].)We note that when k = 1 or k = 2, the expected relation between the Coulomb branch of T N,k and the Higgs branch of T N [Σ 0,k ] breaks down. Neither the 3d nor the 4d theories are CFT's in this case. Nevertheless, the Coulomb branch of T N,k is still a well-defined hyperkähler manifold, in fact a smooth manifold. We will see explicitly that the Coulombbranch chiral rings of T N,k are consistent with k = 1 : M C T * SL(N, C...

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