Temperley-Lieb Immanants

Rhoades, Brendon; Skandera, Mark

doi:10.1007/s00026-005-0268-0

Cited by 28 publications

(33 citation statements)

References 33 publications

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“…We limit ourselves to discussing Theorems 9 and 7 of which we make use in this paper. For detailed exposition of the beautiful results of Rhoades and Skandera we refer reader to the original papers [RS,RS2]. One can also find a (more detailed than here) review in [LPP].…”

Section: Temperley-lieb Immanantsmentioning

confidence: 99%

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On products of sln characters and support containment

Dobrovolska

Pylyavskyy

2007

Journal of Algebra

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Let λ, μ, ν and ρ be dominant weights of sl n satisfying λ + μ = ν + ρ. Let V λ denote the highest weight module corresponding to λ. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for V λ ⊗ V μ to be contained in V ν ⊗ V ρ as sl n -modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible sl n -module which appears in the decomposition of V λ ⊗ V μ does appear in the decomposition of V ν ⊗ V ρ .

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Section: Temperley-lieb Immanantsmentioning

confidence: 99%

“…Rhoades and Skandera [RS2] defined the A product of generators (decomposition) t i 1 · · · t i l in the Temperley-Lieb algebra TL n can be graphically presented by a Temperley-Lieb diagram with n non-crossing strands connecting the vertices 1, . .…”

Section: Temperley-lieb Immanantsmentioning

confidence: 99%

On products of sln characters and support containment

Dobrovolska

Pylyavskyy

2007

Journal of Algebra

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“…(See, e.g., [49 We will use Theorem 5.5 to construct matrices as follows. Fix w = w 1 · · · w n in S n and suppose that for some indices p ≤ q ≤ r, the longest p-decreasing subsequence σ = σ 1 ⊎ · · · ⊎ σ p of w 1 · · · w r has length q.…”

Section: Triangularity Of Transition Matricesmentioning

confidence: 99%

“…Note that span{Imm w (x) | w ∈ S n,1 } is just the one-dimensional space span{det(x)}. The next space has dimension 1 n+1 2n n and was studied in [49], where the elements {Imm w (x) | w ∈ S n,2 } were related to the Temperley-Lieb algebra. For progress on the description of basis elements of span{Imm w (x) | w ∈ S n,3 }, see [46].…”

Section: Filtrations Of the Immanant Spacementioning

confidence: 99%

“…Indeed, we will continue such work begun in [49], [50], [51] by considering a family of nested subsets defined in terms of a partial order on S n which we call iterated dominance. This partial order helps to describe spaces spanned by dual canonical basis elements, their zero sets, and a unitriangular transition matrix relating the bitableau and dual canonical bases.…”

Section: Introductionmentioning

confidence: 99%

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Bitableaux and Zero Sets of Dual Canonical Basis Elements

Rhoades

Skandera

2011

Ann. Comb.

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Abstract. We state new results concerning the zero sets of polynomials belonging to the dual canonical basis of C[x 1,1 , . . . , x n,n ]. As an application, we show that this basis is related by a unitriangular transition matrix to the simpler bitableau basis popularized by Désarménien-Kung-Rota. It follows that spaces spanned by certain subsets of the dual canonical basis can be characterized in terms of products of matrix minors, or in terms of their common zero sets.

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Total positivity for loop groups II: Chevalley generators

Lam

Pylyavskyy

2013

Transformation Groups

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This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequences of braid moves, called a braid limit. We relate this to a partial order, called the limit weak order, on infinite reduced words.The limit semigroup generated by Chevalley generators has a transfinite structure. We prove a form of unique factorization for its elements, in effect reducing their study to infinite products which have the order structure of N. For the latter infinite products, we show that one always has a factorization which matches an infinite Coxeter element.One of the technical tools we employ is a totally positive exchange lemma which appears to be of independent interest. This result states that the exchange lemma (in the context of Coxeter groups) is compatible with total positivity in the form of certain inequalities. 42 9.3. Proof of Lemma 9.2 43 10. Open problems and conjectures 44 References 48

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Temperley-Lieb Immanants

Cited by 28 publications

References 33 publications

On products of sln characters and support containment

On products of sln characters and support containment

Bitableaux and Zero Sets of Dual Canonical Basis Elements

Total positivity for loop groups II: Chevalley generators

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