Transitioning Between Tableaux and Spider Bases for Specht Modules

Im, Mee Seong; Zhu, Jieru

doi:10.1007/s10468-020-10026-6

Cited by 4 publications

(3 citation statements)

References 15 publications

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“…This characterization depends only on their permutation structures, especially their cycle structures. Using this characterization, we prove the results in [RT19, IZ22] concerning the unitriangularity of the transition matrix and a necessary and sufficient condition for additional vanishing entries.…”

Section: Characterization Of Web Permutationsmentioning

confidence: 60%

“…Our second main result, Theorem 3.4, explains how to characterize these permutations in terms of their cycle structures. Using this characterization, we deduce the results in [RT19,IZ22] concerning the unitriangularity of the transition matrix and a necessary and sufficient condition for additional vanishing entries. Furthermore, the characterization leads us to enumerative properties of web permutations.…”

Section: Introduction and The Main Resultsmentioning

confidence: 94%

“…As an application of the characterization, we show that the transition matrix is unitriangular with respect to a certain order on and , and we determine which entries vanish. These are already known due to Russell–Tymoczko [RT19] and Im–Zhu [IZ22].…”

Section: Characterization Of Web Permutationsmentioning

confidence: 88%

See 2 more Smart Citations

A combinatorial model for the transition matrix between the Specht and -web bases

Hwang,

Jang,

2023

Forum of Mathematics, Sigma

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We introduce a new class of permutations, called web permutations. Using these permutations, we provide a combinatorial interpretation for entries of the transition matrix between the Specht and $\operatorname {SL}_2$ -web bases of the irreducible $ \mathfrak {S}_{2n} $ -representation indexed by $ (n,n) $ , which answers Rhoades’s question. Furthermore, we study enumerative properties of these permutations.

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Section: Characterization Of Web Permutationsmentioning

confidence: 60%

Section: Introduction and The Main Resultsmentioning

confidence: 94%

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A combinatorial model for the transition matrix between the Specht and -web bases

Hwang,

Jang,

2023

Forum of Mathematics, Sigma

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A web basis of invariant polynomials from noncrossing partitions

Patrias

Pechenik

Striker

2022

Advances in Mathematics

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The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

Russell

Tymoczko

2020

International Mathematics Research Notices

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Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

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Transitioning Between Tableaux and Spider Bases for Specht Modules

Cited by 4 publications

References 15 publications

A combinatorial model for the transition matrix between the Specht and -web bases

A combinatorial model for the transition matrix between the Specht and -web bases

A web basis of invariant polynomials from noncrossing partitions

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

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