Quantum matrices by paths

Casteels, Karel

doi:10.2140/ant.2014.8.1857

Cited by 11 publications

(19 citation statements)

References 19 publications

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“…However, obtaining a generating set for a given prime is often easier when phrased in terms of the Grassmann necklace. (This follows from [5,7] for O q (M m,n ), and is conjectured for O q (Gr(k, n)). )…”

Section: Introductionmentioning

confidence: 61%

An Algorithm to Construct the Le Diagram Associated to a Grassmann Necklace

Agarwala¹,

Fryer²

2019

Glasgow Math. J.

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Le diagrams and Grassmann necklaces both index the collection of positroids in the nonnegative Grassmannian Gr ≥0 (k, n), but they excel at very different tasks: for example, the dimension of a positroid is easily extracted from its Le diagram, while the list of bases of a positroid is far more easily obtained from its Grassmann necklace. Explicit bijections between the two are therefore desirable. An algorithm for turning a Le diagram into a Grassmann necklace already exists; in this note we give the reverse algorithm.

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Section: Introductionmentioning

confidence: 61%

An Algorithm to Construct the Le Diagram Associated to a Grassmann Necklace

Agarwala¹,

Fryer²

2019

Glasgow Math. J.

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“…Generating sets for

scriptH

‐primes in quantum matrices have been studied extensively (for example, ) under various restrictions on

k

and

q

. In particular, generating sets for all 230

scriptH

‐primes in

{scriptO}_{q} false(M_{3} false)

are computed in for fields of arbitrary characteristic and

q

not a root of unity.…”

Section: A Homeomorphism Between Pspec(ofalse(sl3false)) and Spec(scrmentioning

confidence: 99%

“…As in Example 5.16, upon inverting W 33 and [23|23] we obtain the quantum algebra A 2,4 q (resp. Poisson algebra P 2,4 q , for the same matrix q) on the generators [23|23] ±1 , W 23 , W 32 , W ±1 33 . Three H-primes lie above J, which we list below with their corresponding multiplicative sets E JK :…”

mentioning

confidence: 99%

The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

Fryer

2017

Trans. London Math. Soc.

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A bijection ψ is defined between the prime spectrum of quantum SL3 and the Poisson prime spectrum of SL3, and we verify that both ψ and ψ −1 preserve inclusions of primes, that is, ψ is in fact a homeomorphism between these two spaces. This is accomplished by developing a Poisson analogue of the Brown and Goodearl framework for describing the Zariski topology of spectra of quantum algebras, and then verifying directly that in the case of SL 3 these give rise to identical pictures on both the quantum and Poisson sides. As part of this analysis, we study the Poisson primitive spectrum of O(SL3) and obtain explicit generating sets for all of the Poisson primitive ideals.

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“…The H-primes are used to study the prime spectrum of O q (M m,n ), but are also interesting objects of study in their own right, e.g. [5,6,12,17,24]. A key tool in many of these papers is Cauchon diagrams, which we now define.…”

Section: 2mentioning

confidence: 99%

From Grassmann Necklaces to Restricted Permutations and Back Again

Casteels

Fryer

2017

Algebr Represent Theor

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We study the commutative algebras ZJK appearing in Brown and Goodearl's extension of the H-stratification framework, and show that if A is the single parameter quantized coordinate ring of Mm,n, GLn or SLn, then the algebras ZJK can always be constructed in terms of centres of localizations. The main purpose of the ZJK is to study the structure of the topological space spec(A), which remains unknown for all but a few low-dimensional examples. We explicitly construct the required denominator sets using two different techniques (restricted permutations and Grassmann necklaces) and show that we obtain the same sets in both cases. As a corollary, we obtain a simple formula for the Grassmann necklace associated to a cell of totally nonnegative real m × n matrices in terms of its restricted permutation.The key technique in the study of spec(A) is that of H-stratification, where H is a torus acting rationally on A. Write H-spec(A) for the set of H-primes: the prime ideals of A which are also invariant under the action of H. The H-primes induce a stratification of spec(A), where for each J ∈ H-spec(A) the associated stratum isThis stratification has many nice properties. In particular, each spec J (A) is homeomorphic to the prime spectrum of a commutative Laurent polynomial ring Z(A J ), which can be constructed explicitly as the centre of a localization of A/J [3, Theorem II.2.13]. These algebras Z(A J ) are now reasonably well understood, e.g. [1,25].This approach gives us an excellent picture of the individual strata, but it comes at the cost of losing information about the interactions between primes from different strata. Being able to identify inclusions of prime ideals from different strata is crucial for constructing a complete picture of the topological structure of spec(A), but this information is often surprisingly difficult to obtain: for example, in the setting of quantum matrices only the prime spectra of O q (SL 2 ), O q (GL 2 ), and O q (SL 3 ) ([15], [2], [7] respectively) have been fully described.

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Quantum matrices by paths

Cited by 11 publications

References 19 publications

An Algorithm to Construct the Le Diagram Associated to a Grassmann Necklace

An Algorithm to Construct the Le Diagram Associated to a Grassmann Necklace

The prime spectrum of quantum SL3 and the Poisson prime spectrum of its semiclassical limit

From Grassmann Necklaces to Restricted Permutations and Back Again

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