$q$-Hook formula of Gansner type for a generalized Young diagram

Nakada, Kazuyoshi

doi:10.46298/dmtcs.2684

“…In We conclude this paragraph by recalling from [6] the following remarkable property of Kleshchev-Ram's strongly homogeneous modules in R − mod (see Section 4.4) involving Baumann-Kamnitzer-Knutson's morphism D. It can be essentially viewed as a representationtheoretic reformulation of Nakada's colored hook formula [40] using the identity (5.3). As in [6], we consider the following properties of the flag minors x i 1 , .…”

Section: Equivariant Multiplicitiesmentioning

confidence: 96%

Equivariant multiplicities via representations of quantum affine algebras

Casbi¹,

Lǐ²

2021

Preprint

View full text Add to dashboard Cite

For any simply-laced type simple Lie algebra g and any height function ξ adapted to an orientation Q of the Dynkin diagram of g, Hernandez-Leclerc introduced a certain category C ≤ξ of representations of the quantum affine algebra U q ( g), as well as a subcategory C Q of C ≤ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring C[N] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism D ξ on a torus Y ≤ξ containing the image of K 0 (C ≤ξ ) under the truncated qcharacter morphism. We prove that the restriction of D ξ to K 0 (C Q ) coincides with the morphism D recently introduced by Baumann-Kamnitzer-Knutson in their study of equivariant multiplicities of Mirković-Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov-Reshetikhin modules in C Q , as well as certain results by Brundan-Kleshchev-McNamara on the representation theory of quiver Hecke algebras. This alternative description of D allows us to prove a conjecture by the first author on the distinguished values of D on the flag minors of C[N]. We also provide applications of our results from the perspective of Kang-Kashiwara-Kim-Oh's generalized Schur-Weyl duality. Finally, we use Kashiwara-Kim-Oh-Park's recent constructions to define a cluster algebra A Q as a subquotient of K 0 (C ≤ξ ) naturally containing C[N], and suggest the existence of an analogue of the Mirković-Vilonen basis in A Q on which the values of D ξ may be interpreted as certain equivariant multiplicities.

show abstract

“…In [30], Nakada proposed a generalization of the Peterson-Proctor hook formula. Recall from Section 2.1 that α 1 , .…”

Section: Towards Colored Hook Formulasmentioning

confidence: 99%

“…The set MPathpwq is a finite set in bijection with the set of all reduced expressions of w. We refer to [30,Sections 2,7] for more details. Every term of the sum in the right hand side of Equation ( 6) is equal to 1{N !…”

Section: Towards Colored Hook Formulasmentioning

confidence: 99%

“…We show in Section 5.4 that the normal fan of these bodies essentially describes Qin's dominance order. Finally in Section 6 we prove Theorem 6.3 and conclude with some possible connections with the combinatorics of colored hook formulas studied in [30].…”

mentioning

confidence: 94%

“…. , β S N denote the weights of the cluster variables of every seed S. In [30], Nakada proved a colored hook formula that has a very similar form. However the underlying combinatorics is a priori very different from cluster theory.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Newton-Okounkov bodies for categories of modules over quiver Hecke algebras

Casbi¹

2019

Preprint

View full text Add to dashboard Cite

We show that for a finite-type Lie algebra g, Kang-Kashiwara-Kim-Oh's monoidal categorification Cw of the quantum coordinate ring Aqpnpwqq provides a natural framework for the construction of Newton-Okounkov bodies. In particular, this yields for every seed S of Aqpnpwqq a simplex ∆S of codimension 1 in R lpwq . We exhibit various geometric and combinatorial properties of these simplices by characterizing their rational points, their normal fans, and their volumes. The key tool is provided by the explicit description in terms of root partitions of the determinantial modules of a certain seed in Aqpnpwqq constructed in [13,20]. This is achieved using the recent results of . As an application, we prove an equality of rational functions involving root partitions for cluster variables. It implies an expression of the Peterson-Proctor hook formula in terms of heights of monoidal cluster variables in Cw, suggesting further connections between cluster theory and the combinatorics of fully-commutative elements of Weyl groups.Contents 24 References 27 α 1 `α2 `α3 which is exactly the statement of Theorem 6.3. Remark 6.9. The sums of rational functions on the right hand side of Equations ( 6) and ( 7) are a priori of a very different combinatorial natures. For instance these sums do not have the same number of terms in general. Moreover when specializing the α i to 1, the terms in the right hand side of Equation ( 6) all take the same value 1{N !. On the contrary the value taken by the term indexed by a seed S in Equation ( 7) is essentially the volume of the simplex ∆ S , which is not the same for every seed. However, it turns out (even in more complicated examples) that these two different expressions take rather similar forms. This might suggest closer connections between the combinatorics of fully-commutative elements of Weyl groups and cluster theory. Remark 6.10. Rational functions of the form of Equation (6.3) also appeared in the recent work of Baumann-Kamnitzer-Knutson [1]. They are related with the definition of Duistermaat-Heckmann measures used to compare various bases in A q pnq. In this framework, they prove that the Mirkovic-Vilonen basis and the dual semicanonical basis are not the same.

show abstract

Equivariant multiplicities via representations of quantum affine algebras

Casbi

¹

,

Lǐ

²

2022

Sel. Math. New Ser.

0

View full text Add to dashboard Cite

For any simply-laced type simple Lie algebra $$\mathfrak {g}$$ g and any height function $$\xi $$ ξ adapted to an orientation Q of the Dynkin diagram of $$\mathfrak {g}$$ g , Hernandez–Leclerc introduced a certain category $$\mathcal {C}^{\le \xi }$$ C ≤ ξ of representations of the quantum affine algebra $$U_q(\widehat{\mathfrak {g}})$$ U q ( g ^ ) , as well as a subcategory $$\mathcal {C}_Q$$ C Q of $$\mathcal {C}^{\le \xi }$$ C ≤ ξ whose complexified Grothendieck ring is isomorphic to the coordinate ring $$\mathbb {C}[\textbf{N}]$$ C [ N ] of a maximal unipotent subgroup. In this paper, we define an algebraic morphism $${\widetilde{D}}_{\xi }$$ D ~ ξ on a torus $$\mathcal {Y}^{\le \xi }$$ Y ≤ ξ containing the image of $$K_0(\mathcal {C}^{\le \xi })$$ K 0 ( C ≤ ξ ) under the truncated q-character morphism. We prove that the restriction of $${\widetilde{D}}_{\xi }$$ D ~ ξ to $$K_0(\mathcal {C}_Q)$$ K 0 ( C Q ) coincides with the morphism $$\overline{D}$$ D ¯ recently introduced by Baumann–Kamnitzer–Knutson in their study of equivariant multiplicities of Mirković–Vilonen cycles. This is achieved using the T-systems satisfied by the characters of Kirillov–Reshetikhin modules in $$\mathcal {C}_Q$$ C Q , as well as certain results by Brundan–Kleshchev–McNamara on the representation theory of quiver Hecke algebras. This alternative description of $$\overline{D}$$ D ¯ allows us to prove a conjecture by the first author on the distinguished values of $$\overline{D}$$ D ¯ on the flag minors of $$\mathbb {C}[\textbf{N}]$$ C [ N ] . We also provide applications of our results from the perspective of Kang–Kashiwara–Kim–Oh’s generalized Schur–Weyl duality. Finally, we use Kashiwara–Kim–Oh–Park’s recent constructions to define a cluster algebra $$\overline{\mathcal {A}}_Q$$ A ¯ Q as a subquotient of $$K_0(\mathcal {C}^{\le \xi })$$ K 0 ( C ≤ ξ ) naturally containing $$\mathbb {C}[\textbf{N}]$$ C [ N ] , and suggest the existence of an analogue of the Mirković–Vilonen basis in $$\overline{\mathcal {A}}_Q$$ A ¯ Q on which the values of $${\widetilde{D}}_{\xi }$$ D ~ ξ may be interpreted as certain equivariant multiplicities.

show abstract

$q$-Hook formula of Gansner type for a generalized Young diagram

Cited by 13 publications

References 0 publications

Equivariant multiplicities via representations of quantum affine algebras

Equivariant multiplicities via representations of quantum affine algebras

Newton-Okounkov bodies for categories of modules over quiver Hecke algebras

Equivariant multiplicities via representations of quantum affine algebras

Contact Info

Product

Resources

About