QUASI-DETERMINANTS AND <i>q</i>-COMMUTING MINORS

Lauve, Aaron

doi:10.1017/s0017089510000509

Cited by 4 publications

(4 citation statements)

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“…It is proved in [7] that, in an n × n generic q-matrix, the flag minors with column sets I, J ⊆ [n] quasicommute if and only if the sets I, J are weakly separated. See also [6].) Important properties shown in [7] are that any weakly separated collection C ⊆ 2 [n] has cardinality at most n+1 2 + 1 and that the set of such collections is closed under weak flips (which are defined as for TP-bases above).…”

Section: Introductionmentioning

confidence: 99%

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Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

Данилов

Karzanov

Koshevoy

2010

Advances in Mathematics

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For the ordered set [n] of n elements, we consider the class B n of bases B of tropical Plücker functions on 2 [n] such that B can be obtained by a series of mutations (flips) from the basis formed by the intervals in [n]. We show that these bases are representable by special wiring diagrams and by certain arrangements generalizing rhombus tilings on the n-zonogon. Based on the generalized tiling representation, we then prove that each weakly separated set-system in 2 [n] having maximum possible size belongs to B n , thus answering affirmatively a conjecture due to Leclerc and Zelevinsky. We also prove an analogous result for a hyper-simplex ∆ m n = {S ⊆ [n] : |S| = m}.for any triple i < j < k in [n] and any subset X ⊆ [n] − {i, j, k}. Throughout, for brevity we writeis called a TP-basis, or simply a basis, if the restriction map res : T P n → R B is a bijection. In other words, each TP-function is determined by its values on B, and moreover, values on B can be chosen arbitrarily. 2The first goal of this paper is to characterize B n . We give two characterizations for semi-normal bases: via a bijection to special collections of curves, that we call proper wirings, and via a bijection to certain graphical arrangements, called generalized tilings or, briefly, g-tilings (in fact, these characterizations are interrelated via planar duality). We associate to a proper wiring W (a g-tiling T ) a certain collection of subsets of [n] called its spectrum. It turns out that proper wirings and g-tilings are rigid objects, in the sense that any of these is determined by its spectrum.(More precisely, by a wiring we mean a set of n directed non-self-intersecting curves w 1 , . . . , w n in a region R of the plane homeomorphic to a circle, where each w i begins at a point s i and ends at a point s ′ i , and the points s 1 , . . . , s n , s ′ 1 , . . . , s ′ n are different and occur in this order clockwise in the boundary of R. A special wiring W that we deal with is defined by three axioms (W1)-(W3). Axiom (W1) is standard, it says that W preserves (topologically) under small deformations, i.e., no three wires have a common point, any two wires meet at a finite number of points and they cross, not touch, at each of these points. (W2) says that the common points of w i , w j follow in the opposed orders along these wires. The crucial axiom (W3) says that in the planar graph induced by W , there is a certain bijection between the faces ("chambers") whose boundary is a directed cycle and the regions ("lenses") surrounded by pieces of two wires between their neighboring common points. W is called proper if none of "cyclic" faces is a whole lens. The spectrum of W is the collection of subsets X ⊆ [n] oneto-one corresponding to the "non-cyclic" faces F , where X consists of the elements i such that F "lies on the right" from w i (according to the direction of w i ). When any two wires intersect exactly once, the dual planar graph is realized by a rhombus tiling, and vice versa (for a more general result of this sort, see [5]). In a...

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Section: Introductionmentioning

confidence: 99%

“…It is proved in [7] that, in an n × n generic q-matrix, the flag minors with column sets I, J ⊆ [n] quasicommute if and only if the sets I, J are weakly separated. See also [6]. )…”

mentioning

confidence: 96%

Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

Данилов

Karzanov

Koshevoy

2010

Advances in Mathematics

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“…Since that time, they have proven useful in a number of different settings [GKL + 95, MR04, DFK11]. Specific to the present discussion is the use of quasi-Plücker coordinates to describe coordinate rings for quantum flags and Grassmannians [Lau10].…”

Section: Appendix a Using Euler's Identitymentioning

confidence: 99%

Rational series in the free group and the Connes operator

Lauve¹,

Reutenauer²

2013

Noncommutative Birational Geometry, Representations and Combinatorics

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“…In this paper we take C q = C(q ±1/2 ). The quantized coordinate ring C q [Gr(k, n)] is the subalgebra of the quantum matrix algebra C q [M(k, n)] generated by the quantum Plücker coordinates, see [TT91,GL14,Lau06,Lau10] and Section 3.1. Grabowski and Launois [GL11,GL14] proved that the quantum deformation C q [Gr(k, n)] of C[Gr(k, n)] has a quantum algebra structure.…”

Section: Introductionmentioning

confidence: 99%

Quasi-homomorphisms of quantum cluster algebras

Chang¹,

Huang²,

Lǐ³

2023

Preprint

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In this paper, we study quasi-homomorphisms of quantum cluster algebras, which are quantum analogy of quasi-homomorphisms of cluster algebras introduced by Fraser. Quasi-homomorphisms of quantum cluster algebras appeared in a work by Kimura, Qin, and Wei in 2022 which they called "variation maps".For a quantum Grassmannian cluster algebra C q [Gr(k, n)], we show that there is an associated braid group and each generator σ i of the braid group preserves the quasi-commutative relations of quantum Plücker coordinates and exchange relations of the quantum Grassmannian cluster algebra. We conjecture that σ i also preserves r-term (r ≥ 4) quantum Plücker relations of C q [Gr(k, n)]. Up to this conjecture, we show that σ i is a quasi-automorphism of C q [Gr(k, n)] and the braid group acts on C q [Gr(k, n)].

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QUASI-DETERMINANTS AND q-COMMUTING MINORS

Cited by 4 publications

References 14 publications

Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

Plücker environments, wiring and tiling diagrams, and weakly separated set-systems

Rational series in the free group and the Connes operator

Quasi-homomorphisms of quantum cluster algebras

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