pith. sign in

arxiv: 2605.12880 · v2 · pith:CU7IL5EOnew · submitted 2026-05-13 · 🧮 math.CO · math.RT

Temperley-Lieb Immanants of Ribbon Decomposition Matrices

Son Nguyen , Pavlo Pylyavskyy This is my paper

Pith reviewed 2026-05-20 22:03 UTC · model grok-4.3

classification 🧮 math.CO math.RT
keywords Temperley-Lieb immanantsribbon decomposition matricesSchur positivitydual canonical basisskew Schur functionsimmanantssymmetric functions
0
0 comments X

The pith

Temperley-Lieb immanants are Schur-positive when evaluated on ribbon decomposition matrices.

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

The paper proves that Temperley-Lieb immanants, certain elements of the dual canonical basis, are Schur-positive when evaluated on ribbon decomposition matrices. These matrices supply determinantal formulas for skew Schur functions that generalize several classical identities. A sympathetic reader cares because such positivity results often point to combinatorial models or algebraic structures that explain sign patterns in expansions. The proof covers particular elements and comes with a conjecture extending it to the full basis.

Core claim

Ribbon decomposition matrices give determinantal formulas for skew Schur functions that include as special cases several classical formulas. We prove that certain elements of the dual canonical basis, called Temperley-Lieb immanants, are Schur-positive when evaluated on ribbon decomposition matrices. We conjecture that this positivity holds for all elements of the dual canonical basis.

What carries the argument

Temperley-Lieb immanants evaluated on ribbon decomposition matrices

Load-bearing premise

Ribbon decomposition matrices provide determinantal formulas for skew Schur functions.

What would settle it

Computing the Schur expansion of a Temperley-Lieb immanant on a specific ribbon decomposition matrix and finding a negative coefficient would disprove the positivity.

Figures

Figures reproduced from arXiv: 2605.12880 by Pavlo Pylyavskyy, Son Nguyen.

Figure 3
Figure 3. Figure 3: Skew shape and content The corresponding ribbon decomposition matrix is   1   4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

Ribbon decomposition matrices give determinantal formulas for skew Schur functions that include as special cases the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz formulas. We prove that certain elements of Lusztig's dual canonical basis, called Temperley-Lieb immanants, are Schur-positive when evaluated on ribbon decomposition matrices. We conjecture that this positivity holds for all elements of the dual canonical basis. This is known in the special case of Jacobi-Trudi matrices by a result of Haiman.

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 / 2 minor

Summary. The paper defines ribbon decomposition matrices that yield determinantal expressions for skew Schur functions, encompassing the classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz identities as special cases. It proves that Temperley-Lieb immanants—specific elements of Lusztig's dual canonical basis—are Schur-positive upon evaluation on these matrices. The argument extends Haiman's known positivity result for the Jacobi-Trudi case. A conjecture is stated that the positivity property holds for every element of the dual canonical basis.

Significance. If the central claim is correct, the result supplies a concrete extension of Haiman's theorem to a wider family of determinantal matrices arising from ribbon decompositions. This strengthens evidence for positivity phenomena tied to the Temperley-Lieb algebra action on the dual canonical basis and furnishes an explicit combinatorial setting in which to test the broader conjecture. The direct use of the determinantal formula and the absence of hidden sign assumptions or circular reductions constitute a clear technical contribution to algebraic combinatorics.

minor comments (2)
  1. The manuscript should include at least one fully worked small example (e.g., a 3-ribbon or 4-ribbon matrix) that explicitly computes the Temperley-Lieb immanant expansion and verifies non-negative coefficients, to make the sign-preservation argument more accessible.
  2. Notation for the Temperley-Lieb algebra generators and their action on the dual canonical basis elements should be collected in a single preliminary subsection for quick reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript. We are pleased that the work is viewed as a clear technical contribution extending Haiman's theorem and providing a setting to test the broader conjecture on the dual canonical basis. The recommendation for minor revision is noted, and we will incorporate any such changes in the revised version.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes Schur-positivity of Temperley-Lieb immanants on ribbon decomposition matrices through explicit definitions of the immanants via Temperley-Lieb algebra action and direct verification that expansion coefficients remain non-negative under the given matrix substitution. It relies on the determinantal formulas for skew Schur functions (which generalize classical Jacobi-Trudi, Giambelli, and Lascoux-Pragacz identities) and invokes Haiman's independent prior result only for the special Jacobi-Trudi case. No load-bearing self-citations, self-definitional reductions, fitted inputs renamed as predictions, or ansatz smuggling occur; the central argument supplies its own explicit constructions and coefficient checks without reducing to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background from symmetric function theory and Lusztig's canonical basis constructions with no free parameters, no invented entities, and no ad-hoc axioms introduced to reach the result.

axioms (2)
  • standard math Standard properties of Schur functions, skew Schur functions, and determinantal identities in symmetric function theory.
    Invoked to define the ribbon decomposition matrices and their relation to classical formulas.
  • standard math Existence and basic properties of Lusztig's dual canonical basis and Temperley-Lieb elements within it.
    Used to identify the specific immanants whose positivity is proved.

pith-pipeline@v0.9.0 · 5605 in / 1245 out tokens · 49022 ms · 2026-05-20T22:03:24.630025+00:00 · methodology

Review history (2 revisions) →

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

64 extracted references · 64 canonical work pages

  1. [1]

    Remmel, J. B. and Whitney, R. , TITLE =. J. Algorithms , FJOURNAL =. 1984 , NUMBER =. doi:10.1016/0196-6774(84)90002-6 , URL =

    work page doi:10.1016/0196-6774(84)90002-6 1984

  2. [2]

    Philosophical Transactions of the Royal Society of London

    Littlewood, Dudley Ernest and Richardson, Archibald Read , TITLE =. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character , VOLUME =. 1934 , NUMBER =. doi:10.1098/rsta.1934.0015 , URL =

    work page doi:10.1098/rsta.1934.0015 1934

  3. [3]

    Zelevinsky, A. V. , TITLE =. J. Algebra , FJOURNAL =. 1981 , NUMBER =. doi:10.1016/0021-8693(81)90128-9 , URL =

    work page doi:10.1016/0021-8693(81)90128-9 1981

  4. [4]

    Berenstein, A. D. and Zelevinsky, A. V. , TITLE =. J. Algebraic Combin. , FJOURNAL =. 1992 , NUMBER =. doi:10.1023/A:1022429213282 , URL =

    work page doi:10.1023/a:1022429213282 1992

  5. [5]

    Berenstein, Arkady and Zelevinsky, Andrei , TITLE =. Invent. Math. , FJOURNAL =. 2001 , NUMBER =. doi:10.1007/s002220000102 , URL =

    work page doi:10.1007/s002220000102 2001

  6. [6]

    Knutson, Allen and Tao, Terence , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1999 , NUMBER =. doi:10.1090/S0894-0347-99-00299-4 , URL =

    work page doi:10.1090/s0894-0347-99-00299-4 1999

  7. [7]

    Reiner, Victor and Shimozono, Mark , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1998 , NUMBER =. doi:10.1006/jcta.1997.2841 , URL =

    work page doi:10.1006/jcta.1997.2841 1998

  8. [8]

    Lam, Thomas and Postnikov, Alexander and Pylyavskyy, Pavlo , TITLE =. Amer. J. Math. , FJOURNAL =. 2007 , NUMBER =. doi:10.1353/ajm.2007.0045 , URL =

    work page doi:10.1353/ajm.2007.0045 2007

  9. [9]

    Discrete Comput

    Lam, Thomas and Postnikov, Alexander , TITLE =. Discrete Comput. Geom. , FJOURNAL =. 2007 , NUMBER =. doi:10.1007/s00454-006-1294-3 , URL =

    work page doi:10.1007/s00454-006-1294-3 2007

  10. [10]

    Lie groups, geometry, and representation theory , SERIES =

    Lam, Thomas and Postnikov, Alexander , TITLE =. Lie groups, geometry, and representation theory , SERIES =. 2018 , ISBN =. doi:10.1007/978-3-030-02191-7\_10 , URL =

    work page doi:10.1007/978-3-030-02191-7 2018

  11. [11]

    Dobrovolska, Galyna and Pylyavskyy, Pavlo , TITLE =. J. Algebra , FJOURNAL =. 2007 , NUMBER =. doi:10.1016/j.jalgebra.2006.10.033 , URL =

    work page doi:10.1016/j.jalgebra.2006.10.033 2007

  12. [12]

    Nguyen, Son and Pylyavskyy, Pavlo , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2025 , NUMBER =. doi:10.1093/imrn/rnaf080 , URL =

    work page doi:10.1093/imrn/rnaf080 2025

  13. [13]

    Reiner, Victor and Shimozono, Mark , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1995 , NUMBER =. doi:10.1016/0097-3165(95)90083-7 , URL =

    work page doi:10.1016/0097-3165(95)90083-7 1995

  14. [14]

    Some positive differences of products of

    Bergeron, Fran. Some positive differences of products of. 2004 , ARCHIVEPREFIX =

    work page 2004

  15. [15]

    Fomin, Sergey and Fulton, William and Li, Chi-Kwong and Poon, Yiu-Tung , TITLE =. Amer. J. Math. , FJOURNAL =. 2005 , NUMBER =

    work page 2005

  16. [16]

    Lascoux, Alain and Leclerc, Bernard and Thibon, Jean-Yves , TITLE =. J. Math. Phys. , FJOURNAL =. 1997 , NUMBER =. doi:10.1063/1.531807 , URL =

    work page doi:10.1063/1.531807 1997

  17. [17]

    Okounkov, Andrei , TITLE =. Adv. Math. , FJOURNAL =. 1997 , NUMBER =. doi:10.1006/aima.1997.1622 , URL =

    work page doi:10.1006/aima.1997.1622 1997

  18. [18]

    Pylyavskyy, Pavlo , TITLE =

    work page

  19. [19]

    Rhoades, Brendon and Skandera, Mark , TITLE =. Ann. Comb. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00026-005-0268-0 , URL =

    work page doi:10.1007/s00026-005-0268-0 2005

  20. [20]

    Rhoades, Brendon and Skandera, Mark , TITLE =. J. Algebra , FJOURNAL =. 2006 , NUMBER =. doi:10.1016/j.jalgebra.2005.07.017 , URL =

    work page doi:10.1016/j.jalgebra.2005.07.017 2006

  21. [21]

    Haiman, Mark , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1993 , NUMBER =. doi:10.2307/2152777 , URL =

    work page doi:10.2307/2152777 1993

  22. [22]

    Lusztig, George , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 1990 , NUMBER =. doi:10.2307/1990961 , URL =

    work page doi:10.2307/1990961 1990

  23. [23]

    Duke Math

    Kashiwara, Masaki , TITLE =. Duke Math. J. , FJOURNAL =. 1991 , NUMBER =. doi:10.1215/S0012-7094-91-06321-0 , URL =

    work page doi:10.1215/s0012-7094-91-06321-0 1991

  24. [24]

    Fomin, Sergey and Zelevinsky, Andrei , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2002 , NUMBER =. doi:10.1090/S0894-0347-01-00385-X , URL =

    work page doi:10.1090/s0894-0347-01-00385-x 2002

  25. [25]

    Duke Math

    Kashiwara, Masaki , TITLE =. Duke Math. J. , FJOURNAL =. 1993 , NUMBER =. doi:10.1215/S0012-7094-93-06920-7 , URL =

    work page doi:10.1215/s0012-7094-93-06920-7 1993

  26. [26]

    Berenstein, Arkady and Fomin, Sergey and Zelevinsky, Andrei , TITLE =. Adv. Math. , FJOURNAL =. 1996 , NUMBER =. doi:10.1006/aima.1996.0057 , URL =

    work page doi:10.1006/aima.1996.0057 1996

  27. [27]

    Du, Jie , TITLE =. Bull. London Math. Soc. , FJOURNAL =. 1992 , NUMBER =. doi:10.1112/blms/24.4.325 , URL =

    work page doi:10.1112/blms/24.4.325 1992

  28. [28]

    Du, Jie , TITLE =. J. London Math. Soc. (2) , FJOURNAL =. 1995 , NUMBER =. doi:10.1112/jlms/51.3.461 , URL =

    work page doi:10.1112/jlms/51.3.461 1995

  29. [29]

    Skandera, Mark , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2008 , NUMBER =. doi:10.1016/j.jpaa.2007.09.007 , URL =

    work page doi:10.1016/j.jpaa.2007.09.007 2008

  30. [30]

    Lie theory and geometry , SERIES =

    Lusztig, George , TITLE =. Lie theory and geometry , SERIES =. 1994 , ISBN =. doi:10.1007/978-1-4612-0261-5\_20 , URL =

    work page doi:10.1007/978-1-4612-0261-5 1994

  31. [31]

    Algebraic groups and

    Lusztig, George , TITLE =. Algebraic groups and. 1997 , ISBN =

    work page 1997

  32. [32]

    Localisation de

    Be. Localisation de. C. R. Acad. Sci. Paris S\'. 1981 , NUMBER =

    work page 1981

  33. [33]

    and Kashiwara, M

    Brylinski, J.-L. and Kashiwara, M. , TITLE =. Invent. Math. , FJOURNAL =. 1981 , NUMBER =. doi:10.1007/BF01389272 , URL =

    work page doi:10.1007/bf01389272 1981

  34. [34]

    and Knuth, Donald E

    Bender, Edward A. and Knuth, Donald E. , TITLE =. J. Combinatorial Theory Ser. A , FJOURNAL =. 1972 , PAGES =. doi:10.1016/0097-3165(72)90007-6 , URL =

    work page doi:10.1016/0097-3165(72)90007-6 1972

  35. [35]

    Chmutov, Michael and Glick, Max and Pylyavskyy, Pavlo , TITLE =. J. Comb. Algebra , FJOURNAL =. 2020 , NUMBER =. doi:10.4171/JCA/36 , URL =

    work page doi:10.4171/jca/36 2020

  36. [36]

    Sch\"utzenberger, M. P. , TITLE =. Discrete Math. , FJOURNAL =. 1972 , PAGES =. doi:10.1016/0012-365X(72)90062-3 , URL =

    work page doi:10.1016/0012-365x(72)90062-3 1972

  37. [37]

    Berenstein, A. D. and Kirillov, A. N. , TITLE =. Algebra i Analiz , FJOURNAL =. 1995 , NUMBER =

    work page 1995

  38. [38]

    When is the multiplicity of a weight equal to

    Berenshte. When is the multiplicity of a weight equal to. Funktsional. Anal. i Prilozhen. , FJOURNAL =. 1990 , NUMBER =. doi:10.1007/BF01077330 , URL =

    work page doi:10.1007/bf01077330 1990

  39. [39]

    Discrete Math

    Chiang, Judy Hsin-Hui and Hoang, Anh Trong Nam and Kendall, Matthew and Lynch, Ryan and Nguyen, Son and Przybocki, Benjamin and Xia, Janabel , TITLE =. Discrete Math. , FJOURNAL =. 2024 , NUMBER =. doi:10.1016/j.disc.2024.114068 , URL =

    work page doi:10.1016/j.disc.2024.114068 2024

  40. [40]

    2023 , ARCHIVEPREFIX =

    Nguyen, Son , TITLE =. 2023 , ARCHIVEPREFIX =

    work page 2023

  41. [41]

    1997 , PAGES =

    Fulton, William , TITLE =. 1997 , PAGES =

    work page 1997

  42. [42]

    , TITLE =

    Stanley, Richard P. , TITLE =. [2024] 2024 , PAGES =

    work page 2024

  43. [43]

    Hamel, A. M. and Goulden, I. P. , TITLE =. European J. Combin. , FJOURNAL =. 1995 , NUMBER =. doi:10.1016/0195-6698(95)90002-0 , URL =

    work page doi:10.1016/0195-6698(95)90002-0 1995

  44. [44]

    European J

    Lascoux, Alain and Pragacz, Piotr , TITLE =. European J. Combin. , FJOURNAL =. 1988 , NUMBER =. doi:10.1016/S0195-6698(88)80053-2 , URL =

    work page doi:10.1016/s0195-6698(88)80053-2 1988

  45. [45]

    Skandera, Mark , TITLE =. J. Algebraic Combin. , FJOURNAL =. 2004 , NUMBER =. doi:10.1023/B:JACO.0000047282.21753.ae , URL =

    work page doi:10.1023/b:jaco.0000047282.21753.ae 2004

  46. [46]

    Nguyen, Chau and Nguyen, Son and Woodruff, Dora , journal=. Shuffle

    work page

  47. [47]

    Localisation de g -modules

    Alexandre Be linson and Joseph Bernstein. Localisation de g -modules. C. R. Acad. Sci. Paris S\' e r. I Math. , 292(1):15--18, 1981

    work page 1981

  48. [48]

    Parametrizations of canonical bases and totally positive matrices

    Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. Parametrizations of canonical bases and totally positive matrices. Adv. Math. , 122(1):49--149, 1996

    work page 1996

  49. [49]

    Brylinski and M

    J.-L. Brylinski and M. Kashiwara. Kazhdan- L usztig conjecture and holonomic systems. Invent. Math. , 64(3):387--410, 1981

    work page 1981

  50. [50]

    Canonical bases for irreducible representations of quantum GL _n

    Jie Du. Canonical bases for irreducible representations of quantum GL _n . Bull. London Math. Soc. , 24(4):325--334, 1992

    work page 1992

  51. [51]

    Canonical bases for irreducible representations of quantum GL _n

    Jie Du. Canonical bases for irreducible representations of quantum GL _n . II . J. London Math. Soc. (2) , 51(3):461--470, 1995

    work page 1995

  52. [52]

    Cluster algebras

    Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I . F oundations. J. Amer. Math. Soc. , 15(2):497--529, 2002

    work page 2002

  53. [53]

    Hecke algebra characters and immanant conjectures

    Mark Haiman. Hecke algebra characters and immanant conjectures. J. Amer. Math. Soc. , 6(3):569--595, 1993

    work page 1993

  54. [54]

    A. M. Hamel and I. P. Goulden. Planar decompositions of tableaux and S chur function determinants. European J. Combin. , 16(5):461--477, 1995

    work page 1995

  55. [55]

    On crystal bases of the Q -analogue of universal enveloping algebras

    Masaki Kashiwara. On crystal bases of the Q -analogue of universal enveloping algebras. Duke Math. J. , 63(2):465--516, 1991

    work page 1991

  56. [56]

    Global crystal bases of quantum groups

    Masaki Kashiwara. Global crystal bases of quantum groups. Duke Math. J. , 69(2):455--485, 1993

    work page 1993

  57. [57]

    Ribbon S chur functions

    Alain Lascoux and Piotr Pragacz. Ribbon S chur functions. European J. Combin. , 9(6):561--574, 1988

    work page 1988

  58. [58]

    Canonical bases arising from quantized enveloping algebras

    George Lusztig. Canonical bases arising from quantized enveloping algebras. J. Amer. Math. Soc. , 3(2):447--498, 1990

    work page 1990

  59. [59]

    Shuffle T ableaux, L ittlewood-- R ichardson C oefficients, and S chur L og- C oncavity

    Chau Nguyen, Son Nguyen, and Dora Woodruff. Shuffle T ableaux, L ittlewood-- R ichardson C oefficients, and S chur L og- C oncavity. arXiv preprint arXiv:2506.00349 , 2025

    work page arXiv 2025

  60. [60]

    Temperley- L ieb crystals

    Son Nguyen and Pavlo Pylyavskyy. Temperley- L ieb crystals. Int. Math. Res. Not. IMRN , (7):Paper No. rnaf080, 28, 2025

    work page 2025

  61. [61]

    Temperley- L ieb immanants

    Brendon Rhoades and Mark Skandera. Temperley- L ieb immanants. Ann. Comb. , 9(4):451--494, 2005

    work page 2005

  62. [62]

    Kazhdan- L usztig immanants and products of matrix minors

    Brendon Rhoades and Mark Skandera. Kazhdan- L usztig immanants and products of matrix minors. J. Algebra , 304(2):793--811, 2006

    work page 2006

  63. [63]

    Inequalities in products of minors of totally nonnegative matrices

    Mark Skandera. Inequalities in products of minors of totally nonnegative matrices. J. Algebraic Combin. , 20(2):195--211, 2004

    work page 2004

  64. [64]

    On the dual canonical and K azhdan- L usztig bases and 3412-, 4231-avoiding permutations

    Mark Skandera. On the dual canonical and K azhdan- L usztig bases and 3412-, 4231-avoiding permutations. J. Pure Appl. Algebra , 212(5):1086--1104, 2008

    work page 2008