pith. sign in

arxiv: 2302.12782 · v3 · submitted 2023-02-24 · 🧮 math.RT · math-ph· math.MP

Uncoiled affine Temperley-Lieb algebras and their Wenzl-Jones projectors

Alexis Langlois-R\'emillard , Alexi Morin-Duchesne This is my paper

Pith reviewed 2026-05-24 10:19 UTC · model grok-4.3

classification 🧮 math.RT math-phmath.MP
keywords uncoiled affine Temperley-Lieb algebrasWenzl-Jones projectorssandwich diagramsone-dimensional modulesMarkov tracesChebyshev polynomialsaffine Temperley-Lieb algebras
0
0 comments X

The pith

Uncoiled affine Temperley-Lieb algebras admit explicit Wenzl-Jones idempotents projecting onto each of their one-dimensional modules.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces finite quotients of the infinite-dimensional affine and periodic Temperley-Lieb algebras, which it calls uncoiled versions. These quotients have only finitely many one-dimensional modules. It gives an explicit construction of Wenzl-Jones idempotents that project onto these modules, obtained by adapting the known projectors from ordinary Temperley-Lieb algebras through the sandwich-diagram presentation of the uncoiled algebras. The same framework yields explicit evaluations of Markov traces on the new projectors in terms of Chebyshev polynomials of the first kind.

Core claim

We introduce finite quotients for these algebras, which we term uncoiled affine Temperley-Lieb algebras and uncoiled periodic Temperley-Lieb algebras. The uncoiled algebras all have finitely many one-dimensional modules. We construct a family of Wenzl-Jones idempotents, each of which projects onto one of these one-dimensional modules. Our construction is explicit and uses the similar projectors for the ordinary Temperley-Lieb algebras, as well as the diagrammatic description of the uncoiled algebras in terms of sandwich diagrams. We also discuss the Markov traces for the uncoiled algebras and their evaluations on the newly defined projectors, and find expressions involving Chebyshev polygons

What carries the argument

Uncoiled affine Temperley-Lieb algebras as finite quotients equipped with a sandwich-diagram presentation that transfers Wenzl-Jones projectors from the ordinary Temperley-Lieb case.

If this is right

  • The uncoiled algebras possess only finitely many one-dimensional modules.
  • Markov traces evaluated on the projectors yield expressions in Chebyshev polynomials of the first kind.
  • The uncoiled algebras stand in explicit relation to affine and skew sandwich cellular algebras.
  • Dimensions and defining relations of the uncoiled algebras are finite and explicitly describable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same sandwich-diagram transfer might produce projectors for other finite quotients of diagrammatic algebras.
  • The Chebyshev evaluations could be used to extract closed-form expressions for traces in related statistical-mechanical models.
  • Decomposition of higher-dimensional modules of the uncoiled algebras might now be approachable via these explicit idempotents.

Load-bearing premise

The uncoiled algebras admit a diagrammatic description in terms of sandwich diagrams that permits the explicit transfer and construction of the Wenzl-Jones projectors from the ordinary Temperley-Lieb case.

What would settle it

An explicit matrix computation or relation check showing that the proposed elements are not idempotent or do not annihilate all but one of the one-dimensional modules would disprove the construction.

Figures

Figures reproduced from arXiv: 2302.12782 by Alexi Morin-Duchesne, Alexis Langlois-R\'emillard.

Figure 1
Figure 1. Figure 1: Top left panel: a configuration of the model of bond p [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Affine and periodic Temperley-Lieb algebras are families of diagrammatic algebras that find diverse applications in mathematics and physics. These algebras are infinite dimensional, yet most of their interesting modules are finite. In this paper, we introduce finite quotients for these algebras, which we term uncoiled affine Temperley-Lieb algebras and uncoiled periodic Temperley-Lieb algebras. We study some of their properties, including their defining relations, their description with diagrams, their dimensions, and their relations with affine and skew sandwich cellular algebras. The uncoiled algebras all have finitely many one-dimensional modules. We construct a family of Wenzl-Jones idempotents, each of which projects onto one of these one-dimensional modules. Our construction is explicit and uses the similar projectors for the ordinary Temperley--Lieb algebras, as well as the diagrammatic description of the uncoiled algebras in terms of sandwich diagrams. We also discuss the Markov traces for the uncoiled algebras and their evaluations on the newly defined projectors, and find expressions involving Chebyshev polynomials of the first kind.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript introduces finite quotients of the affine and periodic Temperley-Lieb algebras, called the uncoiled affine and uncoiled periodic Temperley-Lieb algebras. It studies their defining relations, sandwich-diagram presentations, dimensions, and relations to affine and skew sandwich cellular algebras. The algebras are shown to possess only finitely many one-dimensional modules. The paper constructs an explicit family of Wenzl-Jones idempotents projecting onto these modules by adapting the ordinary Temperley-Lieb projectors via the sandwich-diagram description, and it evaluates the Markov traces on these idempotents, obtaining expressions in Chebyshev polynomials of the first kind.

Significance. If the constructions and dimension counts are verified, the work supplies new finite-dimensional quotients of affine TL algebras whose representation theory is tractable, with explicit projectors and trace formulas that could support computations in knot theory and integrable systems. The diagrammatic transfer from the ordinary TL case and the cellular-algebra connections are concrete strengths that make the results usable beyond the present paper.

minor comments (3)
  1. [§3] §3 (defining relations): the additional relations that quotient the affine TL algebra to the uncoiled version should be listed explicitly rather than described only as 'the relations that make the diagrams finite'.
  2. [§5] §5 (projector construction): the verification that the transferred idempotents remain idempotent under the sandwich-diagram multiplication is only sketched; a short direct check for the lowest-rank cases would strengthen the claim.
  3. [Table 1] Table 1 (dimensions): the dimension formula for the uncoiled periodic case is stated without a reference to the cellular-basis count that produces it; adding the relevant cellular-algebra citation or a one-line derivation would clarify the count.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, the assessment of its significance, and the recommendation for minor revision. The report does not list any major comments, so we have no specific points requiring point-by-point response at this stage. We remain available to address any minor revisions or clarifications requested by the editor.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces uncoiled affine and periodic Temperley-Lieb algebras via explicit quotients and a sandwich-diagram presentation, then transfers known Wenzl-Jones projectors from ordinary Temperley-Lieb algebras to obtain idempotents projecting onto the one-dimensional modules. All load-bearing steps are constructive and rely on externally known ordinary TL projectors plus the paper's own diagrammatic relations; no parameter fitting, self-definitional loops, or load-bearing self-citations appear in the derivation chain. The Markov-trace evaluations via Chebyshev polynomials are likewise derived from the explicit projectors rather than presupposed.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Central claim rests on the definition of the uncoiled quotients and the assumption that their sandwich-diagram presentation allows projector transfer; no numerical free parameters appear; axioms are standard facts about Temperley-Lieb algebras and cellular algebras; the uncoiled algebras themselves are the main new defined objects.

axioms (1)
  • standard math Standard diagrammatic relations and cellular structure of ordinary Temperley-Lieb algebras
    The construction re-uses known projectors from the ordinary case.
invented entities (2)
  • Uncoiled affine Temperley-Lieb algebras no independent evidence
    purpose: Finite quotients of affine TL algebras with finitely many 1D modules
    Newly introduced in the paper via quotient construction.
  • Uncoiled periodic Temperley-Lieb algebras no independent evidence
    purpose: Finite quotients of periodic TL algebras with finitely many 1D modules
    Newly introduced in the paper via quotient construction.

pith-pipeline@v0.9.0 · 5727 in / 1366 out tokens · 41794 ms · 2026-05-24T10:19:14.709128+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages · 24 internal anchors

  1. [1]

    Baxter, Exactly Solved Models in Statistical Mechanics , Academic Press (1982)

    R.J. Baxter, Exactly Solved Models in Statistical Mechanics , Academic Press (1982)

    work page 1982

  2. [2]

    Belletˆ ete, A.M

    J. Belletˆ ete, A.M. Gainutdinov, J.L. Jacobsen, H. Sale ur, T.S. Tavares, Topological defects in lattice models and affine Temperley–Lieb algebra , Comm. Math. Phys. 400 (2023) 1203–1254, arXiv:1811.02551 [hep-th]

    work page arXiv 2023

  3. [3]

    On the computation of fusion over the affine Temperley-Lieb algebra

    J. Belletˆ ete, Y. Saint-Aubin, On the computation of fusion over the affine Temperley–Lieb alg ebra, Nucl. Phys. B937 (2018) 333–370, arXiv:1802.03575 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  4. [4]

    Birman, Braids, links and mapping class groups , Ann

    J. Birman, Braids, links and mapping class groups , Ann. Math. Stud. 82 (1975) 228p

    work page 1975

  5. [5]

    Birman, H

    J. Birman, H. Wenzl, Braids, link polynomials and a new algebra , Trans. Amer. Math. Soc. 313 (1989) 249–273

    work page 1989

  6. [6]

    Brauer, On algebras which are connected with the semisimple continu ous groups , Ann

    R. Brauer, On algebras which are connected with the semisimple continu ous groups , Ann. Math. 38 (1937) 857–872

    work page 1937

  7. [7]

    $p$-Jones-Wenzl idempotents

    B. Burrull, N. Libedinsky, P. Sentinelli, p-Jones-Wenzl idempotents, Adv. Math. 352 (2019) 246– 264, arXiv:1902.00305 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  8. [8]

    Nested cobordisms, Cyl-objects and Temperley-Lieb algebras

    M.E. Calle, R.S. Hoekzema, L. Murray, N. Pacheco-Tallaj , C. Rovi, S. Sridhar-Shapiro, Nested cobordisms, Cyl-objects and Temperley–Lieb algebras , arXiv:2403.01067v1 [math.AT]

    work page internal anchor Pith review Pith/arXiv arXiv

  9. [9]

    Colmenarejo, R

    L. Colmenarejo, R. Orellana, F. Saliola, A. Schilling, M . Zabrocki, An insertion algorithm on multiset partitions with applications to diagram algebr as, J. Algebra 557 (2020) 97–128, arXiv:1905.02071 [math.CO]. 51

    work page arXiv 2020

  10. [10]

    Cramp´ e, L

    N. Cramp´ e, L. Poulain d’Andecy, L. Vinet, M. Zaimi, Askey–Wilson Braid Algebra and Centralizer of Uq(sl2), Ann. Henri Poincar´ e24 (2023) 1897–1922, arXiv:2206.11150 [math.RT]

    work page arXiv 2023

  11. [11]

    Cramp´ e, L

    N. Cramp´ e, L. Vinet, M. Zaimi, Temperley–Lieb, Birman–Murakami–Wenzl and Askey– Wilson Algebras and Other Centralizers of Uq(sl2), Ann. Henri Poincar´ e 22 (2021) 3499–3528, arXiv:2008.04905 [math.QA]

    work page arXiv 2021

  12. [12]

    di Francesco, H

    P. di Francesco, H. Saleur, J.B. Zuber, Relations between the Coulomb Gas Picture and Conformal Invariance of Two-Dimensional Critical Models , J. Stat. Phys. 49 (1987) 57–79

    work page 1987

  13. [13]

    The two-color Soergel calculus

    B. Elias, The two-color Soergel calculus , Compos. Math. 152 (2016) 327–398, arXiv:1308.6611 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  14. [14]

    Soergel bimodules for universal Coxeter groups

    B. Elias, N. Libedinsky, Indecomposable Soergel bimodules for universal Coxeter grou ps, Trans. Amer. Math. Soc. 369 (2017) 3883–3910, arXiv:1401.2467 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    Elias, S

    B. Elias, S. Makisumi, U. Thiel, G. Williamson, Introduction to Soergel bimodules, Springer Nature (2020)

    work page 2020

  16. [16]

    On representations of affine Temperley-Lieb algebras, II

    K. Erdmann, R.M. Green, On representations of affine Temperley–Lieb algebras II , Pac. J. Math. 191 (1999) 243–274, arXiv:math/9811017 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  17. [17]

    On the affine Temperley-Lieb algebras

    C.K. Fan, R.M. Green, On the affine Temperley–Lieb algebras , J. London. Math. Soc. 60 (1999) 366–380, arXiv:q-alg/9706003

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  18. [18]

    Frenkel, M.G

    I.B. Frenkel, M.G. Khovanov, Canonical bases in tensor products and graphical calculus fo rUq(sl2), Duke Math. J. 87 (1997) 409–480

    work page 1997

  19. [19]

    Gainutdinov, J.L

    A.M. Gainutdinov, J.L. Jacobsen, H. Saleur, A fusion for the periodic Temperley–Lieb algebra and its continuum limit , J. High Energy Phys. (2018) 117, arXiv:1712.07076 [hep-th]

    work page arXiv 2018

  20. [20]

    The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk Logarithmic Conformal Field Theory at c=0

    A.M. Gainutdinov, N. Read, H. Saleur, R. Vasseur, The periodic sl(2|1) alternating spin chain and its continuum limit as a bulk logairthmic conformal field theory at c = 0, J. High Energy Phys. (2015) 114, arXiv:1409.0167 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  21. [21]

    Fusion and braiding in finite and affine Temperley-Lieb categories

    A.M. Gainutdinov, H. Saleur, Fusion and braiding in finite and affine Temperley–Lieb categor ies, arXiv:1606.04530 [math.QA]

    work page internal anchor Pith review Pith/arXiv arXiv

  22. [22]

    Lattice fusion rules and logarithmic operator product expansions

    A.M. Gainutdinov, R. Vasseur, Lattice fusion rules and logarithmic operator product expa nsions, Nucl. Phys. B868 (2013) 223–270, arXiv:1203.6289 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  23. [23]

    Goodman, H

    F. Goodman, H. Wenzl, The Temperley–Lieb algebra at roots of unity , Pac. J. Math. 161 (1993) 307–334

    work page 1993

  24. [24]

    Graham, G.I

    J.J. Graham, G.I. Lehrer, Cellular algebras , Invent. Math. 123 (1996) 1–34

    work page 1996

  25. [25]

    Graham, G.I

    J.J. Graham, G.I. Lehrer, The representation theory of affine Temperley–Lieb algebras , Enseign. Math. 44 (1998) 173–218

    work page 1998

  26. [26]

    Graham, G.I

    J.J. Graham, G.I. Lehrer, Cellular algebras and diagram algebras in representation th eory, Adv. Stud. Pure Math. 40 (2004) 141–173

    work page 2004

  27. [27]

    Green, On representations of affine Temperley–Lieb algebras , Algebra and modules II, CMS Conference Proceedings, 24 (1998) 245–261 arXiv:2301.12367 [math.RT]

    R.M. Green, On representations of affine Temperley–Lieb algebras , Algebra and modules II, CMS Conference Proceedings, 24 (1998) 245–261 arXiv:2301.12367 [math.RT]. 52

    work page arXiv 1998

  28. [28]

    Commuting Families in Temperley-Lieb Algebras

    T. Halverson, M. Mazzocco, A. Ram, Commuting families in Hecke and Temperley-Lieb algebras , Nagoya Math. J. 195 (2009) 125–152, arXiv:0710.0596 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  29. [29]

    Partition Algebras

    T. Halverson, A. Ram, Partition algebras , Eur. J. Comb. 26 (2005) 869–921, arXiv:math/0401314 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  30. [30]

    Ikhlef, A

    Y. Ikhlef, A. Morin-Duchesne, Fusion in the periodic Temperley–Lieb algebra and connectiv ity operators of loop models , SciPost Phys. 12 (2022) 030, arXiv:2105.10240 [math-ph]

    work page arXiv 2022

  31. [31]

    Ikhlef, A

    Y. Ikhlef, A. Morin-Duchesne, Fusion of irreducible modules in the periodic Temperley–Lie b algebra, SciPost Phys. 17 (2024) 132, arXiv:2312.14837 [math-ph]

    work page arXiv 2024

  32. [32]

    Jacobsen, Conformal field theory applied to loop models , In A.J

    J.L. Jacobsen, Conformal field theory applied to loop models , In A.J. Guttmann, editor, Polygons, polyominoes and polycubes, Springer Verlag, Lect. Notes Ph ys. 775 (2009) 347–424

    work page 2009

  33. [33]

    The $p$-Canonical Basis for Hecke Algebras

    L.T. Jensen, G. Williamson, The p-canonical basis for Hecke algebras , Categorification and Higher Representation Theory, Vol. 683, Contemp. Math. Amer. Math. Soc. (2017) 333–361 arXiv:1510.01556 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  34. [34]

    Jones, Index for subfactors , Invent

    V.F.R. Jones, Index for subfactors , Invent. Math. 72 (1983) 1–25

    work page 1983

  35. [35]

    Jones, A polynomial invariant for knots via von Neumann algebras , Bull

    V.F.R. Jones, A polynomial invariant for knots via von Neumann algebras , Bull. Amer. Math. Soc. (N.S.), 12.1 (1985) 103–111

    work page 1985

  36. [36]

    Jones, The Potts model and the symmetric group , in Subfactors (Kyuzeso 1993), World Sci

    V.F.R. Jones, The Potts model and the symmetric group , in Subfactors (Kyuzeso 1993), World Sci. Publ., River Edge, NJ (1994) 259–267

    work page 1993

  37. [37]

    V.F.R Jones, A quotient of the affine Hecke algebra in the Brauer algebra , Enseign. Math. 40 (1994) 313–344

    work page 1994

  38. [38]

    Kauffman, S.L

    L.H. Kauffman, S.L. Lins, Temperley–Lieb Recoupling Theory and Invariants of 3-manifolds, Princeton University Press (1994)

    work page 1994

  39. [39]

    Kazhdan, G

    D. Kazhdan, G. Lusztig, Representations of Coxeter groups and Hecke algebras , Invent. Math. 53 (1979) 165–184

    work page 1979

  40. [40]

    Khovanov, R

    M. Khovanov, R. Sazdanovic, Diagrammatic categorification of the Chebyshev polynomials of the second kind, J. Pure App. Algebra 225 (2021) 106592, arXiv:2003.11664 [math.RT]

    work page arXiv 2021

  41. [41]

    Koenig, C

    S. Koenig, C. Xi, Affine cellular algebras , Adv. Math. 229 (2012) 139–182

    work page 2012

  42. [42]

    Langlois-R´ emillard, Y

    A. Langlois-R´ emillard, Y. Saint-Aubin,The representation theory of seam algebras, SciPost Phys. 8 (2020) 019, arXiv:1909.03499 [math-ph]

    work page arXiv 2020

  43. [43]

    Levy, Algebraic structure of translation-invariant spin- 1 XXZ and q-Potts quantum chains , Phys

    D. Levy, Algebraic structure of translation-invariant spin- 1 XXZ and q-Potts quantum chains , Phys. Rev. Lett. 67 (1991) 1971–1974

    work page 1991

  44. [44]

    Martin, Potts Models and Related Problems in Statistical Mechanics , World Scientific (1991)

    P. Martin, Potts Models and Related Problems in Statistical Mechanics , World Scientific (1991)

    work page 1991

  45. [45]

    Martin, The structure of the partition alegbras , J

    P. Martin, The structure of the partition alegbras , J. Algebra 183 (1996) 319–358

    work page 1996

  46. [46]

    Martin, H

    P. Martin, H. Saleur, On an algebraic approach to higher dimensional statistical mechanics, Comm. Math. Phys. 158 (1993) 155–190. 53

    work page 1993

  47. [47]

    Martin, H

    P. Martin, H. Saleur, Algebras in higher-dimensional statistical mechanics — th e exceptional partition (mean field) algebras , Lett. Math. Phys. 30 (1994) 179–185

    work page 1994

  48. [48]

    Martin, R.A

    S. Martin, R.A. Spencer, ( ℓ,p )-Jones-Wenzl idempotents , J. Algebra 603 (2022) 41–60, arXiv:2102.08205 [math.RT]

    work page arXiv 2022

  49. [49]

    [Mun57] W.D

    A. Mathas, D. Tubbenhauer, Cellularity for weighted KLR W algebras of types B, A(2), D(2), J. London Math. Soc. 2 (2022) 1–43, arXiv:2201.01998 [math.RT]

    work page arXiv 2022

  50. [50]

    Morin-Duchesne, A

    A. Morin-Duchesne, A. Kl¨ umper, P.A. Pearce,Modular covariant torus partition functions of dense A (1) 1 and dilute A (2) 2 loop models, in preparation

    work page

  51. [51]

    Modular invariant partition function of critical dense polymers

    A. Morin-Duchesne, P.A. Pearce, J. Rasmussen, Modular invariant partition function of critical dense polymers, Nucl. Phys. B874 (2013) 312–357, arXiv:1303.4895 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  52. [52]

    Fusion hierarchies, T-systems and Y-systems of logarithmic minimal models

    A. Morin-Duchesne, P.A. Pearce, J. Rasmussen, Fusion hierarchies, T -systems, and Y -systems of logarithmic minimal models , J. Stat. Mech. (2014) P05012, arXiv:1401.7750 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  53. [53]

    Boundary algebras and Kac modules for logarithmic minimal models

    A. Morin-Duchesne, J. Rasmussen, D. Ridout, Boundary algebras and Kac modules for logarithmic minimal models, Nucl. Phys. B899 (2015) 677–769, arXiv:1503.07584 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  54. [54]

    A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

    A. Morin-Duchesne, Y. Saint-Aubin, A homomorphism between link and XXZ modules over the periodic Temperley–Lieb algebra , J. Phys. A: Math. Theor. 46 (2013) 285207, arXiv:1203.4996 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  55. [55]

    Murakami, The Kauffman polynomial of links and representation theory , Osaka J

    J. Murakami, The Kauffman polynomial of links and representation theory , Osaka J. Math. 24 (1987) 745–758

    work page 1987

  56. [56]

    Pasquier, H

    V. Pasquier, H. Saleur, Common structures between finite systems and conformal field t heories through quantum groups , Nucl. Phys. B330 (1990) 523–556

    work page 1990

  57. [57]

    Logarithmic Minimal Models

    P.A. Pearce, J. Rasmussen, J.B. Zuber, Logarithmic minimal models, J. Stat. Mech. (2006) P11017, arXiv:hep-th/0607232

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  58. [58]

    Pinet, Y

    T. Pinet, Y. Saint-Aubin, Spin chains as modules over the affine Temperley–Lieb algebra , Algebr. Represent. Theor. (2022), arXiv:2205.02649 [math.RT]

    work page arXiv 2022

  59. [59]

    Extremal weight projectors

    H. Queffelec, P. Wedrich, Extremal weight projectors , Math. Res. Lett. 25 (2018) 1911–1936, arXiv:1701.02316 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  60. [60]

    Fusion Algebras of Logarithmic Minimal Models

    J. Rasmussen, P.A. Pearce, Fusion algebras of logarithmic minimal models , J. Phys. A: Math. Theor. 40 (2007) 13711, arXiv:0707.3189v2 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  61. [61]

    N. Read, H. Saleur, Associative-algebraic approach to logarithmic conformal field theories , Nucl. Phys. B777 (2007) 316–351, arXiv:hep-th/0701117

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  62. [62]

    Eigenvalue amplitudes of the Potts model on a torus

    J.F. Richard, J.L. Jacobsen, Eigenvalue amplitudes of the Potts model on a torus , Nucl. Phys. B769 (2007) 256–274, arXiv:math-ph/0608055 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  63. [63]

    Standard Modules, Induction and the Temperley-Lieb Algebra

    D. Ridout, Y. Saint-Aubin, Standard modules, induction and the structure of the Temper ley–Lieb algebra, Adv. Theor. Math. Phys. 18 (2014) 957–1041, arXiv:1204.4505 [math-ph]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  64. [64]

    Sklyanin, Boundary conditions for integrable quantum systems , J

    E.K. Sklyanin, Boundary conditions for integrable quantum systems , J. Phys. A: Math. Gen. 21 (1988) 2375–2389. 54

    work page 1988

  65. [65]

    Spencer, Modular Valenced Temperley–Lieb Algebras, Rocky Mt J

    R.A. Spencer, Modular Valenced Temperley–Lieb Algebras, Rocky Mt J. Math. 53 1 (2023) 177– 208, arXiv:2108.10011 [math.RT]

    work page arXiv 2023

  66. [66]

    Sutton, D

    L. Sutton, D. Tubbenhauer, P. Wedrich, J. Zhu, SL 2 tilting modules in the mixed case , Sel. Math. New Ser. 29 (2023) 39, arXiv:2105.07724 [math.RT]

    work page arXiv 2023

  67. [67]

    percolation

    H. Temperley, E. Lieb, Relations between the “percolation ” and “colouring” probl em and other graph-theoretical problems associated with regular plana r lattices: Some exact results for the “percolation ” problem, Proc. Roy. Soc. London Ser. A322 (1971) 251–280

    work page 1971

  68. [68]

    Sandwich cellularity and a version of cell theory

    D. Tubbenhauer, Sandwich cellularity and a version of cell theory , Rocky Mountain J. Math. 54 no. 6 (2024) 1733–1773, arXiv:2206.06678 [math.RT]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  69. [69]

    Tubbenhauer, P

    D. Tubbenhauer, P. Vaz, Handlebody diagram algebras , Rev. Mat. Iberoam. 39 no. 3 (2022) 845– 896, arXiv:2105.07049 [math.QA]

    work page arXiv 2022

  70. [70]

    Webster, G

    B. Webster, G. Williamson, The geometry of Markov traces , Duke Math. J. 160 no. 2 (2011) 401–419, arXiv:0911.4494 [math.AG]

    work page arXiv 2011

  71. [71]

    Wenzl, Hecke algebras of type An and subfactors , Invent

    H. Wenzl, Hecke algebras of type An and subfactors , Invent. Math. 92 (1988) 349–384

    work page 1988

  72. [72]

    Westbury, The representation theory of the Temperley–Lieb Algebras , Math

    B. Westbury, The representation theory of the Temperley–Lieb Algebras , Math. Zeit. 219 (1995) 539–565. 55

    work page 1995