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arxiv: 1906.08740 · v1 · pith:RAWHRS3Nnew · submitted 2019-06-20 · 🧮 math.CO · math.RT

Explicit Combinatorial Formulas for Some Irreducible Characters of the GL_ktimes mathbb{S}_n-module of multivariate diagonal harmonics

Nancy Wallace This is my paper

Pith reviewed 2026-05-25 19:21 UTC · model grok-4.3

classification 🧮 math.CO math.RT
keywords diagonal harmonicsSchur functionscombinatorial formulaspathsirreducible charactersGL_kS_nmultivariate
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The pith

A new path object T_{n,s} expresses some irreducible characters of multivariate diagonal harmonics directly as sums of Schur functions.

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 formulas for certain irreducible components of the GL_k × S_n-module of multivariate diagonal harmonics. It does so by defining a new family of paths T_{n,s} that index those components, turning the character into a direct sum of Schur functions. The same approach supplies Schur-function expressions in the q and t variables for special cases of the operators ∇(e_n), ∇^r(e_n) and Δ'_{e_k}(e_n). It also recasts the adjoint dual Pieri rule on these characters as a statement about paths. Readers interested in representation theory of symmetric groups and diagonal harmonics would gain concrete, computable expressions where only abstract descriptions existed before.

Core claim

The central claim is that the character of certain irreducible GL_k × S_n-submodules inside the multivariate diagonal harmonics equals the sum, indexed by the newly defined paths T_{n,s}, of the corresponding Schur functions; the same paths also furnish explicit Schur expressions in q and t for special cases of ∇(e_n), ∇^r(e_n) and Δ'_{e_k}(e_n) and interpret the adjoint dual Pieri rule combinatorially.

What carries the argument

The path object T_{n,s}, which indexes the relevant irreducible components so that the character becomes a sum of Schur functions.

If this is right

  • The character of special cases of ∇(e_n) can be written explicitly as a sum of Schur functions in the q and t variables.
  • Analogous explicit Schur formulas hold for ∇^r(e_n) and Δ'_{e_k}(e_n) in the indicated cases.
  • The adjoint dual Pieri rule applied to these characters admits a direct path interpretation.
  • The formulas give a combinatorial rule for extracting the irreducible components that appear in the decomposition.

Where Pith is reading between the lines

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

  • The path indexing may simplify the calculation of multiplicities when these characters appear inside larger modules.
  • Similar path constructions could be tested on other symmetric-function operators or on modules for larger groups.
  • The formulas make it feasible to compute the dimension or the q,t-degree of each indexed component for moderate n and s.

Load-bearing premise

The paths T_{n,s} are defined so that the sum of their associated Schur functions equals the character of the irreducible component they index.

What would settle it

Compute the GL_k × S_n-character for small concrete values such as n=3, k=2, s=1 by direct representation-theoretic methods and check whether it equals the sum of Schur functions over the paths T_{3,1}.

Figures

Figures reproduced from arXiv: 1906.08740 by Nancy Wallace.

Figure 3
Figure 3. Figure 3: τ ∈ SYT(42211); des(τ ) = 4, maj(τ ) = 19 Des(τ ) = {2, 4, 5, 8}, We end this section by recalling the definition of the Gaussian polynomials, n k [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: e ⊥ 2 s4311 = s3211 + s331 + s421 + s43 4. Lifting to multivariate formulas The following results gives us a way to lift to an alternating sum. We will show later how to obtain positive sums from these [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: T7,2 • • • • • [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparing path ending with a North step in C n−1 j+k+1 and with an east step in C n−1 j+k [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Concatenation of a path of Tl+2,0 and a path of C n−2−l j−1 • • j n − 1 [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: gives an example for r = 1 and n = 4. One might want to take note that for n ≤ 4 there are only hooks thus hE4,4, e4i|hooks = hE4,4, e4i. This is why we can state in the introduction that the equation of theorem 3.2.5 in [BCP18] is a specialization of Equation (1) using a different basis. The elements of T2,0: area(γ) : ht(γ) : hook(γ) : shook(γ) : • • 3 2 6, 1 0 s6 , • • 2 1 4, 1 1 s41 , • • 1 1 3, 1 1 s… view at source ↗
Figure 13
Figure 13. Figure 13: The map e ⊥ 2+ send the path γ = NENEENEE ∈ T10,0 to the path NNENEE ∈ T10,des(τ ′) , with Des(τ ′ ) = {6, 8} . The map e ⊥ k−(γ) is defined in a similar way. For γ ∈ Tn,0, let us consider the prefix of γ ending with the k − 1-th east step. We denote by p1, . . . , pk−1 the integers such that pi is the number of north steps before the i-th east step. Let h be the number of east steps before the first nort… view at source ↗
Figure 14
Figure 14. Figure 14: ). Note that we take out the path En−2 because it is associated to the Schur function s1n and e ⊥ k (s1n ) has only one term. ht • • • • • • • • • • γ = ht maj(τ ′ ) • • • • • • • • • • e ⊥ 2−(γ) = [PITH_FULL_IMAGE:figures/full_fig_p017_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparing the path NNEENEE = N2Eγ˜ ∈ T10,1 and the path ˜γ ∈ T9,3. Same height ht(N2Eγ˜) = ht(˜γ) though area(N2Eγ˜) = area(˜γ) + 3. • • • • • • • • • • r k n − 2 Tn,k Tn−r,k+1 [PITH_FULL_IMAGE:figures/full_fig_p021_15.png] view at source ↗
Figure 18
Figure 18. Figure 18: An example of the map Ω1 1 when n = 7, k = 1 and j = 1. With NNEEN ∈ T (1) 7,0,3 . created are all distinct values of {1, . . . , n − 1}. Hence, Φk associates γ to a unique tableau of shape (k + 1, 1 n−k−1 ) consequently the maps, Φk is well defined. Let γ, π ∈ T E n,0 be such that Φk(γ) = Φk(π). By the previous paragraph the height of the path determines the shape of the tableau thus γ and π are of same … view at source ↗
Figure 19
Figure 19. Figure 19: An example of the map β2 when n = 7 and d = 2. The image β2(τ ) is in T7,0,3. Lemma 13. The maps βd are well defined bijections. Moreover, for all τ we have maj(τ ) = area(βd(τ )) + ht(βd(τ )) + 1 [PITH_FULL_IMAGE:figures/full_fig_p025_19.png] view at source ↗
read the original abstract

We give an explicit combinatorial formula for some irreducible components of $GL_k\times \mathbb{S}_n$-modules of multivariate diagonal harmonics. To this end we introduce a new path combinatorial object $T_{n,s}$ allowing us to give the formula directly in terms of Schur functions. This paper also contains formulas written in terms of Schur functions in the $q$ and $t$ variables for special cases of $\nabla(e_n)$, $\nabla^r(e_n)$ and $\Delta'_{e_k}(e_n)$. We also give an interpretation in term of path to the adjoint dual Pieri rule applied on these $GL_k\times \mathbb{S}_n$-characters.

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 a new combinatorial path object T_{n,s} to index certain irreducible components of the GL_k × S_n-module of multivariate diagonal harmonics, yielding explicit formulas for these components expressed as sums of Schur functions. It additionally supplies Schur-function expressions (in q,t variables) for special cases of the operators ∇(e_n), ∇^r(e_n) and Δ'_{e_k}(e_n), together with a path-based interpretation of the adjoint dual Pieri rule applied to the same characters.

Significance. If the claimed indexing and equalities hold, the work supplies explicit, combinatorial expressions for selected irreducible characters that are otherwise difficult to compute explicitly. The construction is internally self-contained, relying only on the definition of T_{n,s} and standard combinatorial operations on paths and Schur functions rather than external conjectures; this is a genuine strength for a paper in algebraic combinatorics.

minor comments (3)
  1. The definition and enumeration of the path set T_{n,s} should be accompanied by at least one fully worked small example (e.g., n=3 or n=4) that shows both the paths and the resulting Schur-function sum, to make the indexing claim immediately verifiable.
  2. Notation for the special-case operators ∇^r and Δ'_{e_k} is introduced without an explicit reminder of their standard definitions; a one-sentence recall in the relevant section would improve readability.
  3. The manuscript should state clearly whether the formulas for the special cases of ∇(e_n) etc. are derived as corollaries of the main T_{n,s} construction or obtained by independent arguments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its self-contained combinatorial approach, and recommendation of minor revision. No specific major comments appear in the report, so we have no individual points requiring response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity; self-contained combinatorial construction

full rationale

The paper defines a new path object T_{n,s} and states that the character equals the indicated sum of Schur functions over this indexing set. Both the definition of T_{n,s} and the claimed equality are supplied internally as explicit combinatorial rules with no fitted parameters, no load-bearing self-citations, and no reduction of the target identity to its own inputs by construction. The argument remains a direct enumeration and does not invoke external conjectures or prior results whose verification depends on the present work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 1 invented entities

Abstract only; no explicit free parameters, background axioms, or invented entities beyond the new object T_{n,s} are visible.

invented entities (1)
  • T_{n,s} path object no independent evidence
    purpose: To index irreducible components and produce the Schur-function formula for the characters
    New combinatorial object defined in the paper for the stated purpose.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

  1. [1]

    Bergeron, C

    N. Bergeron, C. Ceballos, and V. Pilaud. Hopf dreams, 2018

    work page 2018

  2. [2]

    Bergeron

    F. Bergeron. Structural properties of the ( gl_k S _n )-modules of multivariate diagonal harmonics polynomials. In preperation

    work page

  3. [3]

    Algebraic combinatorics and coinvariant spaces

    Fran c ois Bergeron. Algebraic combinatorics and coinvariant spaces . CMS Treatises in Mathematics. Canadian Mathematical Society, Ottawa, ON; A K Peters, Ltd., Wellesley, MA, 2009

    work page 2009

  4. [4]

    Multivariate diagonal coinvariant spaces for complex reflection groups

    Fran c ois Bergeron. Multivariate diagonal coinvariant spaces for complex reflection groups. Adv. Math. , 239:97--108, 2013

    work page 2013

  5. [5]

    Bergeron and A

    F. Bergeron and A. M. Garsia. Science fiction and M acdonald's polynomials. In Algebraic methods and q -special functions ( M ontr\'eal, QC , 1996) , volume 22 of CRM Proc. Lecture Notes , pages 1--52. Amer. Math. Soc., Providence, RI, 1999

    work page 1996

  6. [6]

    Bergeron, A

    F. Bergeron, A. M. Garsia, M. Haiman, and G. Tesler. Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions. Methods Appl. Anal. , 6(3):363--420, 1999. Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part III

    work page 1999

  7. [7]

    Higher trivariate diagonal harmonics via generalized T amari posets

    Fran c ois Bergeron and Louis-Fran c ois Pr\' e ville-Ratelle. Higher trivariate diagonal harmonics via generalized T amari posets. J. Comb. , 3(3):317--341, 2012

    work page 2012

  8. [8]

    Carlson and A

    E. Carlson and A. Mellit. A proof of the schuffle conjecture. J. Amer. Math. Soc. , 31(8), 2018

    work page 2018

  9. [9]

    A. M. Garsia and M. Haiman. A remarkable q,t - C atalan sequence and q - L agrange inversion. J. Algebraic Combin. , 5(3):191--244, 1996

    work page 1996

  10. [10]

    Gorsky, A

    E. Gorsky, A. Negut, and J. Rasmussen. Flag hilbert schemes, colored projectors and khovanov-rozansky homology, 2016

    work page 2016

  11. [11]

    J. Haglund. A proof of the q,t - S chr\"oder conjecture. Int. Math. Res. Not. , 2004(11):525--560, 2004

    work page 2004

  12. [12]

    Vanishing theorems and character formulas for the H ilbert scheme of points in the plane

    Mark Haiman. Vanishing theorems and character formulas for the H ilbert scheme of points in the plane. Invent. Math. , 149(2):371--407, 2002

    work page 2002

  13. [13]

    Haglund, M

    J. Haglund, M. Haiman, N. Loehr, J. B. Remmel, and A. Ulyanov. A combinatorial formula for the character of the diagonal coinvariants. Duke Math. J. , 126(2):195--232, 2005

    work page 2005

  14. [14]

    Hogancam

    M. Hogancam. Khovnov-rozansky homology and higher catalan sequences, 2017

    work page 2017

  15. [15]

    Haglund, B

    J. Haglund, B. Rhoades, and M. Shimozono. Ordered set partitions generalized coinvariant algebras and the delta conjecture. Adv. Math. , 329(1):851--915., 2018

    work page 2018

  16. [16]

    Triply-graded link homology and H ochschild homology of S oergel bimodules

    Mikhail Khovanov. Triply-graded link homology and H ochschild homology of S oergel bimodules. Internat. J. Math. , 18(8):869--885, 2007

    work page 2007

  17. [17]

    I. G. Macdonald. Symmetric functions and H all polynomials . Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications

    work page 1995

  18. [18]

    C. Mellit. Toric braids and (m,n)-parking functions, 2016

    work page 2016

  19. [19]

    Knot homology and sheaves on the H ilbert scheme of points on the plane

    Alexei Oblomkov and Lev Rozansky. Knot homology and sheaves on the H ilbert scheme of points on the plane. Selecta Math. (N.S.) , 24(3):2351--2454, 2018

    work page 2018

  20. [20]

    schurize

    N. Wallace. An attempt to "schurize" the schr\"oder paths and bijections into tableaux. In preperation

    work page

  21. [21]

    N. Wallace. A new path combinatorial object bijected into known combinatorial objects. In preperation

    work page

  22. [22]

    N. Wallace. Explicit combinatorial formulas for some irreducible characters of the gl_k S _n -module of multivariate diagonal harmonics. Séminaire Lotharingien de Combinatoire: Proceedings of the 31st Conference on Formal Power Series and Algebraic Combinatorics (Ljubljana) , page 12., 2019

    work page 2019