pith. sign in

arxiv: 2604.06479 · v2 · submitted 2026-04-07 · 🧮 math.CO · math.AT· math.RT

Stability and ribbon bases for the rank-selected homology of geometric lattices

Patricia Hersh , Sheila Sundaram This is my paper

Pith reviewed 2026-05-13 01:14 UTC · model grok-4.3

classification 🧮 math.CO math.ATmath.RT
keywords geometric latticesrank-selected homologyribbon basesBoolean latticepartition latticeWhitney homologystability bounds
0
0 comments X p. Extension

The pith

Geometric lattices admit new ribbon bases for their rank-selected homology, yielding sharp stability bounds for Boolean and partition lattices.

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

The paper constructs explicit combinatorial bases, called ribbon bases, for the homology groups that arise when only certain ranks are selected inside geometric lattices. These bases are shown to work for every geometric lattice and for both the ordinary rank-selected homology and the Whitney version. The construction is then applied to the Boolean lattice and the partition lattice to obtain exact thresholds beyond which the action of symmetries on these homology groups no longer changes with the size of the lattice. A reader would care because the result turns an infinite family of algebraic objects into a finite computation once the stability threshold is passed, and supplies a concrete combinatorial model that replaces abstract existence arguments.

Core claim

Ribbon bases exist for the rank-selected homology and rank-selected Whitney homology of any geometric lattice; these bases are linearly independent and span the modules, and they imply sharp uniform stability bounds on the representation-theoretic behavior of the Boolean lattice for all rank sets and of the partition lattice for arbitrary rank sets.

What carries the argument

Ribbon bases, a combinatorially defined collection of homology classes built from the ranked structure and exchange properties of the lattice, that serve as an explicit spanning and independent set for the modules in question.

If this is right

  • The stability bound for the partition lattice is sharp and holds for every possible choice of rank set.
  • The Boolean lattice satisfies uniform stability with an explicit numerical threshold that depends only on the selected ranks.
  • The same ribbon construction supplies bases for the Whitney homology modules of any geometric lattice.
  • Once the bases are available, the algebraic invariants of large lattices reduce to calculations on finitely many small examples.

Where Pith is reading between the lines

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

  • The ribbon construction may extend to other ranked posets that satisfy the semimodular and exchange axioms, allowing similar explicit bases beyond geometric lattices.
  • These bases could be used to define multiplication or other operations on the homology that respect the lattice structure.
  • Stability thresholds obtained this way might guide numerical experiments that search for analogous stabilization in homology of non-geometric posets.

Load-bearing premise

The proposed ribbon elements must actually be linearly independent and span the full homology module for every geometric lattice.

What would settle it

For a concrete small geometric lattice, compute the rank-selected homology dimension directly and test whether the proposed ribbon vectors are linearly dependent or fail to reach that dimension; the same check applied to a partition lattice with a chosen rank set would refute the claimed stability bound if the dimension changes after the predicted threshold.

Figures

Figures reproduced from arXiv: 2604.06479 by Patricia Hersh, Sheila Sundaram.

Figure 1
Figure 1. Figure 1: Left to right: the connected skew shape (5, 4, 3, 3)/(2, 2, 1) and the ribbon Rib(3, 1, 4, 2) A ribbon (also called a border strip in [29, 37], and a skew hook in [43]), is a connected skew shape that does not contain a 2 by 2 square of four boxes. We specify a ribbon by its sequence of row lengths (r1, r2, . . . , rk) from bottom to top, and we denote this by Rib(r1, r2, . . . , rk). See [PITH_FULL_IMAGE… view at source ↗
Figure 2
Figure 2. Figure 2: A small example of a Young symmetrizer annihilating a ribbon homology element. The ribbon entries are an NBC independent set in Π4. Corollary 6.1 below to deduce Conjecture 11.3 of [22]. In Section 6.1, we will prove that WHS(Πn) cannot stabilize earlier than 4 max S − |S| + 1. Then we lay the groundwork for proving stability and also verify our desired stability bound for WHS(Πn) in special cases such as … view at source ↗
Figure 3
Figure 3. Figure 3: A subset L of the first row of T, indicated by the boxes labelled X, as in the proof of Lemma 6.21. to be the subset of the first row of T consisting of the leftmost λ1(u) letters in that first row. See [PITH_FULL_IMAGE:figures/full_fig_p049_3.png] view at source ↗
read the original abstract

This paper analyzes the representation theoretic stability, in the sense of Thomas Church and Benson Farb, of the rank-selected homology of the Boolean lattice and the partition lattice, proving sharp uniform representation stability bounds in both cases. It proves a conjecture of the first author and Reiner by giving the sharp stability bound for general rank sets for the partition lattice. Along the way, a new homology basis sharing useful features with the polytabloid basis for Specht modules is introduced for the rank-selected homology and for the rank-selected Whitney homology of any geometric lattice, resolving an old open question of Bj\"orner. These bases give a matroid theoretic analogue of Specht modules.

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 proves sharp uniform representation stability bounds, in the sense of Church and Farb, for the rank-selected homology of the Boolean lattice and the partition lattice. It establishes the sharp stability bound for general rank sets in the partition lattice, thereby proving a conjecture of the first author and Reiner. It also constructs explicit ribbon bases for the rank-selected homology and rank-selected Whitney homology of arbitrary geometric lattices; these bases resolve an open question of Björner and furnish a matroid-theoretic analogue of the polytabloid basis for Specht modules.

Significance. If the claimed proofs and basis constructions hold, the results are significant: they supply the first sharp stability bounds for these families, resolve a long-standing question on bases for geometric-lattice homology, and introduce combinatorial objects that parallel Specht modules in a matroid setting. The explicit, matroid-theoretic nature of the ribbon bases is likely to enable further computations and generalizations in combinatorial representation theory.

minor comments (2)
  1. The abstract refers to 'ribbon bases' without a preliminary definition or comparison to existing bases (e.g., the NBC bases of Björner); a short introductory paragraph or diagram in §1 would help readers unfamiliar with the construction.
  2. Notation for the rank-selected homology modules H_k(Δ, S) and the action of S_n is introduced gradually; a consolidated notation table or list of conventions at the end of the introduction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on sharp representation stability bounds and the construction of ribbon bases, as well as their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external axioms and prior frameworks

full rationale

The paper constructs explicit ribbon bases for rank-selected homology and Whitney homology of arbitrary geometric lattices by leveraging the standard ranked, semimodular, and exchange properties of geometric lattices, then applies the Church-Farb representation stability framework to obtain sharp uniform bounds for the Boolean and partition lattices while proving a prior conjecture. No step reduces a claimed prediction or basis to a fitted input by construction, no load-bearing uniqueness theorem is imported solely via self-citation, and the central results rest on combinatorial axioms and external stability theory rather than re-deriving inputs from outputs. The abstract and claims exhibit no self-definitional loops or renaming of known results as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions from lattice theory and representation theory; no free parameters are introduced or fitted, and no new entities are postulated without construction.

axioms (2)
  • domain assumption Geometric lattices are ranked, semimodular posets satisfying the exchange axiom
    Invoked to define rank-selected homology and to guarantee the existence of the new bases across all such lattices
  • domain assumption Representation stability in the sense of Church-Farb applies to the symmetric group actions on the homology modules
    Used to formulate and prove the uniform stability bounds

pith-pipeline@v0.9.0 · 5403 in / 1679 out tokens · 50282 ms · 2026-05-13T01:14:31.536909+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.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

45 extracted references · 45 canonical work pages

  1. [1]

    Barcelo, On the action of the symmetric group on the free Lie algebra and the partition lattice,J

    H. Barcelo, On the action of the symmetric group on the free Lie algebra and the partition lattice,J. Combinatorial Theory, Ser. A55(1990), no. 1, 93–129

    work page 1990

  2. [2]

    Barcelo and N

    H. Barcelo and N. Bergeron, The Orlik-Solomon algebra on the partition lattice and the free Lie algebra. J. Combinatorial Theory, Ser. A55(1990), no. 1, 80–92

    work page 1990

  3. [3]

    Cohen-Macaulay ordered sets.J

    Kenneth Baclawski. Cohen-Macaulay ordered sets.J. Algebra, 63(1):226–258, 1980

    work page 1980

  4. [4]

    Shellable and Cohen–Macaulay partially ordered sets.Trans

    Anders Bj¨ orner. Shellable and Cohen–Macaulay partially ordered sets.Trans. Amer. Math. Soc., 260(1):159–183, 1980

    work page 1980

  5. [5]

    On the homology of geometric lattices.Algebra Universalis, 14(1):107–128, 1982

    Anders Bj¨ orner. On the homology of geometric lattices.Algebra Universalis, 14(1):107–128, 1982

    work page 1982

  6. [6]

    Bj¨ orner

    A. Bj¨ orner. The homology and shellability of matroids and geometric lattices. InMatroid applications, Encyclopedia Math. Appl.40,Cambridge Univ. Press, Cambridge, (1992), 226-283

    work page 1992

  7. [7]

    On lexicographically shellable posets.Trans

    Anders Bj¨ orner and Michelle Wachs. On lexicographically shellable posets.Trans. Amer. Math. Soc., 277(1):323–341, 1983

    work page 1983

  8. [8]

    Anders Bj¨ orner and Michelle L. Wachs. Shellable nonpure complexes and posets. I.Trans. Amer. Math. Soc., 348(4):1299–1327, 1996

    work page 1996

  9. [9]

    Buch, A Littlewood-Richardson rule for theK-theory of Grassmannians.Acta Mathematica189 (2002), 37–78

    A. Buch, A Littlewood-Richardson rule for theK-theory of Grassmannians.Acta Mathematica189 (2002), 37–78

    work page 2002

  10. [10]

    Church, Homological stability for configuration spaces of manifolds.Invent

    T. Church, Homological stability for configuration spaces of manifolds.Invent. Math.188(2012), no. 2, 465–504

    work page 2012

  11. [11]

    Church, J

    T. Church, J. Ellenberg and B. Farb, Representation stability in cohomology and asymptotics for families of varieties over finite fields. InAlgebraic Topology: Applications and New Directions, Contemp. Math. 620, Amer. Math. Soc., Providence, (2014), 1–54

    work page 2014

  12. [12]

    Church, J

    T. Church, J. S. Ellenberg, and B. Farb, FI-modules and stability for representations of symmetric groups.Duke Math. J.,164(2015), 1833–1910

    work page 2015

  13. [13]

    Church and B

    T. Church and B. Farb, Representation theory and homological stability.Adv. Math.245(2013), 250– 314. 55

    work page 2013

  14. [14]

    Curtis and I

    C. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, Pure and Applied MathematicsXI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London 1962

    work page 1962

  15. [15]

    Davidson and P

    R. Davidson and P. Hersh, A lexicographic shellability characterization of geometric lattices.J. Combin. Theory Ser. A,123(2014), 8–13

    work page 2014

  16. [16]

    D. S. Dummit and R. M. Foote, Abstract algebra, 3rd ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004

    work page 2004

  17. [17]

    Fulton,Young Tableaux, London Mathematical Society Student Texts,35, Cambridge University Press, 1997

    W. Fulton,Young Tableaux, London Mathematical Society Student Texts,35, Cambridge University Press, 1997

    work page 1997

  18. [18]

    Hanlon, The fixed point partition lattices,Pacific J

    P. Hanlon, The fixed point partition lattices,Pacific J. Math.96(1981), no. 2, 319–341

    work page 1981

  19. [19]

    Hemmer, Stable decompositions for some symmetric group characters arising in braid group coho- mology.J

    D.J. Hemmer, Stable decompositions for some symmetric group characters arising in braid group coho- mology.J. Combin. Theory Ser. A118(2011), 1136–1139

    work page 2011

  20. [20]

    Hanlon and P

    P. Hanlon and P. Hersh, Multiplicity of the trivial representation in rank-selected homology of the partition lattice,J. Algebra266(2003), no. 2, 521–538

    work page 2003

  21. [21]

    Hersh, Lexicographic shellability for balanced complexes,J

    P. Hersh, Lexicographic shellability for balanced complexes,J. Algebraic Combinatorics17(2003), no. 3, 225–254

    work page 2003

  22. [22]

    Hersh and V

    P. Hersh and V. Reiner, Representation stability for cohomology of configuration spaces inR d.IMRN, 2017(2017), no. 4, 1433–1486

    work page 2017

  23. [23]

    G. D. James, The representation theory of the symmetric groups, Lecture Notes in Mathematics682. Springer-Verlag 1978

    work page 1978

  24. [24]

    Joyal, Foncteurs analytiques et esp` eces de structures

    A. Joyal, Foncteurs analytiques et esp` eces de structures. InCombinatoire ´ enum´ erative (Montreal, Que., 1985), Lecture Notes in Math.1234, pages 126–159, Springer, Berlin, 1986

    work page 1985

  25. [25]

    Knutson, E

    A. Knutson, E. Miller and A. Yong, Tableau complexes,Israel J. Math.,163(2008), 317–343

    work page 2008

  26. [26]

    Knutson, E

    A. Knutson, E. Miller and A. Yong, Gr¨ obner geometry of vertex decompositions and of flagged tableaux. J. Reine Angew. Math.(Crelle’s Journal),630(2009), 1–31

    work page 2009

  27. [27]

    G. I. Lehrer and L. Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes.J. Algebra104no. 2 (1986), 410–424

    work page 1986

  28. [28]

    Marberg and Kam Hung Tong, Crystals for set-valued decomposition tableaux,Algebr

    E. Marberg and Kam Hung Tong, Crystals for set-valued decomposition tableaux,Algebr. Comb.8 (2025), 857–896

    work page 2025

  29. [29]

    Macdonald, Symmetric functions and Hall polynomials, 2nd edition

    I.G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition. Oxford Mathematical Mono- graphs, Oxford 1995

    work page 1995

  30. [30]

    Orlik and L

    P. Orlik and L. Solomon, Combinatorics and topology of complements of hyperplanes.Invent. Math.56 (1980) no. 2, 167–189

    work page 1980

  31. [31]

    Pechenik and T

    O. Pechenik and T. Scrimshaw, K-theoretic crystals for set-valued tableaux of rectangular shapes,Algebr. Comb.5(2022), 515–536

    work page 2022

  32. [32]

    Sagan and S

    B. Sagan and S. Sundaram, Ordered set partition posets, arXiv:2506.2335

    work page arXiv

  33. [33]

    Solomon, A decomposition of the group algebra of a finite Coxeter group.J

    L. Solomon, A decomposition of the group algebra of a finite Coxeter group.J. Algebra9(1968), 220–239

    work page 1968

  34. [34]

    Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966

    E.H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966

    work page 1966

  35. [35]

    Stanley, Finite lattices and Jordan-H¨ older sets.Algebra Universalis,4(1974), 361–371

    R.P. Stanley, Finite lattices and Jordan-H¨ older sets.Algebra Universalis,4(1974), 361–371

    work page 1974

  36. [36]

    Stanley, Enumerative Combinatorics, Vol

    R.P. Stanley, Enumerative Combinatorics, Vol. 1. With a foreword by Gian-Carlo Rota. Corrected reprint of the 1986 original. Cambridge Studies in Advanced Mathematics,49.Cambridge University Press, Cambridge, 1997. xii+325 pp

    work page 1986

  37. [37]

    Stanley, Enumerative Combinatorics, Vol

    R.P. Stanley, Enumerative Combinatorics, Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics,62.Cambridge University Press, Cambridge, 1999. xii + 581 pp

    work page 1999

  38. [38]

    Stanley, Some aspects of groups acting on finite posets.J

    R.P. Stanley, Some aspects of groups acting on finite posets.J. Combin. Theory, Ser. A32(1982), no. 2, 132–161

    work page 1982

  39. [39]

    Sundaram and V

    S. Sundaram and V. Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces.Trans. Amer. Math. Soc.349(1997), no. 4, 1389–1420

    work page 1997

  40. [40]

    Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice.Adv

    S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice.Adv. Math.104(1994), no. 2, 225–296

    work page 1994

  41. [41]

    Wachs, A basis for the homology of thed-divisible partition lattice,Adv

    M. Wachs, A basis for the homology of thed-divisible partition lattice,Adv. Math.117(1996), no. 2, 294–318. 56

    work page 1996

  42. [42]

    Wachs, On the (co)homology of the partition lattice and the free Lie algebra.Disc

    M. Wachs, On the (co)homology of the partition lattice and the free Lie algebra.Disc. Math.197(1998), no. 1-3, 287–319

    work page 1998

  43. [43]

    M. Wachs. Poset topology: tools and applications. InGeometric combinatorics, volume 13 ofIAS/Park City Math. Ser., pages 497–615. Amer. Math. Soc., Providence, RI, 2007

    work page 2007

  44. [44]

    Welsh, Matroid Theory, L.M.S

    D.J.A. Welsh, Matroid Theory, L.M.S. Monographs, No. 8, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976, xi+433 pp

    work page 1976

  45. [45]

    Wilson,F I W –modules and stability criteria for representations of classical Weyl groups.J

    J. Wilson,F I W –modules and stability criteria for representations of classical Weyl groups.J. Algebra 420(2014), 269–332. Department of Mathematics, University of Oregon, Eugene, OR 97403 Email address:plhersh@uoregon.edu Department of Mathematics, University of Minnesota, Minneapolis, MN 55455 Email address:shsund@umn.edu

    work page 2014