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Configuration spaces and peak representations

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Pith's one-line read The paper proves that the peak-idempotent representations of the symmetric group are exactly the even cohomology of the configuration space of n points in RP^2 × R, where they decompose as direct sums of Thrall's higher Lie characters.

desk verdict A strong, original paper with a real but fixable hypothesis gap: Theorem 1.1(i) and (iii) are stated under char(k)>2 but their proof uses a semisimplicity theorem that requires |S±_n| invertible. read the letter →

arxiv 2508.09898 v1 pith:ZZYNJBLH submitted 2025-08-13 math.CO math.RT

Marcelo Aguiar , Sarah Brauner , Victor Reiner This is my paper · ORCID
classification math.COmath.RT MSC 05E1014N2052C3555R80
keywords peakalgebraconfigurationspaceshigherLiecharactersEulerianidempotentsdescentVarchenko-Gelfandringsymmetricgrouprepresentationshyperplanearrangements
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a cohomological interpretation for the peak representations of the symmetric group: for a field k with characteristic larger than n, the representation generated by the k-th peak idempotent is isomorphic to the cohomology $H^{{2k}}$ of the configuration space Z_n of n ordered points in $RP^{2}$ × R. It proves that this cohomology is the regular representation of S_n as an ungraded S_n-module, vanishes except in degrees divisible by 4, and that each even-degree piece decomposes as the direct sum of Thrall's higher Lie characters Lie_λ over partitions λ of n with exactly n−k odd parts. Consequently the Betti numbers count permutations by number of odd cycles, and the paper derives Hilbert series, equivariant branching rules, and a connection to simple Jordan elements. The result extends the known type A and B stories, where Eulerian idempotent representations are cohomology of configuration spaces of types A and B, to the peak algebra.

What carries the argument

The load-bearing objects are the peak idempotents $E^{P_n}_k$ in the peak algebra $P_n \subset kS_n$, obtained by forgetting signs from the type B Eulerian idempotents $E^{S^\pm_n}_k$; the configuration space $Z_n = Y_n/\mathbb{Z}_2^n$, whose cohomology is identified with the $\mathbb{Z}_2^n$-fixed subalgebra of $H^*Y_n$; and the Pairing Lemma 5.2, which bijects the squarefree $T$-basis of $H^*X_n$ with the $\mathbb{Z}_2^n$-invariant monomials in the $V$-presentation of $H^*Y_n$. This bijection yields the $S_n$-equivariant isomorphism $\mathrm{gr}(Y_n)^{\mathbb{Z}_2^n} \cong H^*X_n$, which, together with the known identification of flat-orbit summands with $\mathrm{Lie}_\lambda$, produces th

What would settle it

Work over $\mathbb{Q}$ and compute the Poincaré polynomial of $\mathrm{Conf}_4(\mathbb{RP}^2 \times \mathbb{R})$, for example via the Galois covering $Y_4 \to Z_4$ with known Betti numbers for $Y_4$; Theorem 1.2 predicts $1 + 14t + 9t^2$ (with $t$ counting $H^{4k}$), so any other answer would refute it. More cheaply, verify the recursion $H_n = H_{n-1} + t(n-1)^2 H_{n-2}$ for $n=3,4$ against a direct topological computation.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: whenever char(k) > n, for each even k, $(kS_n)E^{P_n}_{n-k} \cong \bigoplus_{\lambda \vdash n,\ \mathrm{odd}(\lambda)=n-k} \mathrm{Lie}_\lambda \cong H^{2k}(Z_n)$, with $Z_n = \mathrm{Conf}_n(\mathbb{RP}^2 \times \mathbb{R})$. The proof passes through a second, finer statement: after diagonalizing the $\mathbb{Z}_2^n$-action on a Varchenko–Gelfand presentation of $H^*Y_n$, the $\mathbb{Z}_2^n$-fixed subalgebra of the associated graded ring is isomorphic, as an $S_n$-module, to the type A cohomology $H^*X_n$, and this isomorphism matches the bigrading with a decomposition into higher Lie characters indexed by odd and even parts.

Load-bearing premise

The arguments require the field characteristic to be larger than n (and larger than 2), so the group algebras of $S_n$, $S^\pm_n$, and $\mathbb{Z}_2^n$ are semisimple; if that semisimplicity fails, the filtrations cannot be replaced by their associated gradeds and the stated isomorphisms are not established.

Editorial extensions

If this is right

  • For k even, the peak representation $(kS_n)E^{P_n}_{n-k}$ is isomorphic to $H^{2k}(Z_n)$, so every peak-idempotent representation is realized topologically.
  • The Betti number of $H^{2k}(Z_n)$ equals the number of permutations of n with n−k odd cycles, giving a closed-form count and a Hilbert series satisfying the recursion $H_n = H_{n-1} + tq(n-1)(1+q(n-2))H_{n-2}$.
  • The cohomology $H^i(Z_n)$ vanishes unless $i \equiv 0 \bmod 4$, and $E^{P_n}_k = 0$ unless $k \equiv n \bmod 2$, so the peak algebra grading has a parity constraint.
  • The finer primitive peak idempotents $E^{P_n}_\mu$ vanish unless $\mu$ is an odd partition and $n-|\mu|$ is even; the non-vanishing ones form a complete primitive system for $P_n$.
  • The associated graded of $H^*Z_n$ decomposes into $\mathrm{Lie}_\lambda$ with both length and odd-part conditions, refining the cohomological decomposition and recovering the simple Jordan element representation $V_n(-1)$ as the $u_i$-free filtration piece.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: because the proof is semisimple in nature, the asserted isomorphisms likely fail over small characteristic; computing the mod-p Betti numbers of $\mathrm{Conf}_n(\mathbb{RP}^2 \times \mathbb{R})$ for $p \le n$ would provide a concrete test of how the characteristic hypothesis is needed.
  • Editorial inference: the ungraded $S_n$-isomorphism $H^*Z_n \cong H^*X_n$ suggests there may be a graded or filtered version placing the two cohomologies in a common framework, which the paper does not address.
  • Editorial inference: the Betti-number statistic "number of odd cycles" already appears in derangement and Tsetlin-library contexts; the configuration-space realization may give a geometric model for higher Lie characters and a potential new route toward Thrall's problem of decomposing $\mathrm{Lie}_\lambda$ into irreducibles.
  • Editorial inference: the connection to simple Jordan elements suggests the bigrading on $\mathrm{gr}(Y_n)^{\mathbb{Z}_2^n}$ could be interpreted as a Jordan degree, and that q-analogues of the branching rule might correspond to deformations of the Jordan bracket.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the peak algebra P_n inside the group algebra of the symmetric group and the representations generated by its peak idempotents E^{P_n}_k. The main object is the configuration space Z_n = Y_n / Z_2^n, where Y_n is the ordered Z_2-orbit configuration space of n points in R^3; topologically Z_n ≅ Conf_n(RP^2 × R). The paper proves that the total cohomology H^*Z_n is the regular representation kS_n (Theorem 1.1(i)), that the even-degree pieces H^{2k}Z_n are isomorphic to (kS_n)E^{P_n}_{n-k} (Theorem 1.1(ii)), and that cohomology vanishes unless the degree is divisible by 4 (Theorem 1.1(iii)). The central Theorem 1.2 identifies (kS_n)E^{P_n}_{n-k} with a direct sum of Thrall's higher Lie characters Lie_λ over partitions λ with odd(λ)=n-k, and also identifies a refined bigraded component of gr(Y_n)^{Z_2^n} with the sum of Lie_λ satisfying both ℓ(λ)=n-ℓ and odd(λ)=n-k. The proof strategy is to pass from the known type B isomorphism H^*Y_n ≅ kS^±_n to the Z_2^n-fixed subalgebra, diagonalize the Z_2^n action, use a combinatorial Pairing Lemma to identify a monomial basis of the fixed space with the type A nbc-basis, and then use the identification X_λ ≅ Lie_λ. The paper also derives Hilbert series, equivariant Hilbert series, branching rules, and a connection to simple Jordan elements.

Significance. If the main theorems are correct, this is a substantial and elegant contribution. It gives the first cohomological interpretation of peak representations, completes the parallel with the type A and type B Eulerian idempotents, and identifies the peak-idempotent representations with explicit direct sums of higher Lie characters. The statements are precise and the combinatorial backbone—especially the Pairing Lemma and the monomial-basis identifications—is original and well executed. The paper also provides explicit Hilbert-series and branching-rule consequences that are easily testable and likely to be useful. The main theorems are not derived by circular reasoning and no fitted parameters appear. However, one hypothesis gap in Theorem 1.1(i) and its consequences must be repaired before the results can be regarded as fully established.

major comments (2)
  1. [Section 3, Eq. (3.1), isomorphism (b)] Theorem 1.1(i) is asserted for char(k)>2, but its proof invokes Theorem 2.5 with W=S^±_n. Theorem 2.5 explicitly requires |W| ∈ k^×, i.e. 2^n n! invertible in k. For any n≥3 and any prime p with 2<p≤n, the hypothesis char(k)>2 holds but |S^±_n| is divisible by p. For example n=3 and k=F_3 give char(k)>2 but |S^±_3|=48 not invertible. Thus the isomorphism H^*Y_n ≅ kS^±_n used in step (b), and hence the claimed regular-representation conclusion H^*Z_n ≅ kS_n, is not established as stated. This is load-bearing: Theorem 5.1 and Proposition 6.5 use the dimension n!=dim_k H^*Z_n obtained from Theorem 1.1(i), and Theorem 1.1(iii) (vanishing mod 4) uses Theorem 5.1. The authors should either strengthen Theorem 1.1(i) and the derived statements to char(k)>n, or prove and cite a strengthening of Theorem 2.5 that holds for these specific Eilenberg–MacLane spaces without the full semisimplicity hypo
  2. [Theorem 5.1 / Proposition 6.5] The proof that the monomials in V∩Q Q form a k-basis for gr(Y_n)^{Z_2^n} relies on the dimension count dim_k (Y_n)^{Z_2^n}=n!, which in turn is justified by Theorem 1.1(i). Since Theorem 1.1(i) is not proven under char(k)>2 as noted above, the basis theorem and all later consequences that use it under the weaker hypothesis—including Theorem 1.1(iii) and Proposition 6.5—are unsupported as written. This is not an independent gap but a downstream consequence; it should be resolved together with the preceding issue.
minor comments (4)
  1. [Section 2.5, Eq. (2.10)] The display says φ : Sol(S^±_n) ↠ Sol(S_n). This must be a typo: according to the paper's own definition and [1], the image of Sol(S^±_n) under the forget-sign map is the peak algebra P_n, not all of Sol(S_n).
  2. [Section 7, Corollary 7.1] The notation 'Z_n' is used for the group Z_2^n in a few places (e.g. 'gr(Y_n)^{Z_n}'), which conflicts with the space Z_n. The intended group Z_2^n should be written consistently.
  3. [Section 9.2, after Eq. (9.11)] The branching-rule notation '↓ ↑ ↑' is a little compressed; a one-sentence explanation of which Sn and S_{n-1} are involved in the double induction would improve readability.
  4. [Throughout] The hypotheses on the field are stated carefully in Section 2.6, but because Theorem 1.1(i) and (iii) are later used at char(k)>2 while their proofs require semisimplicity of kS^±_n, the hypothesis summary should be updated once the gap is fixed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main isomorphisms are proved from prior independent theorems and internal combinatorial lemmas.

full rationale

The derivation chain is self-contained against external benchmarks. Theorem 1.1(i)-(ii) follows from the transfer isomorphism H*(Y_n/Γ) ≅ (H*Y_n)^Γ, Moseley's Theorem 2.5 giving H*Y_n ≅ kS±_n, Brauner's prior theorem (1.5) H^{2k}Y_n ≅ (kS±_n)E^{S±_n}_{n-k}, and Lemma 3.1, a general fixed-point/quotient idempotent isomorphism proven in the paper. The peak idempotents are defined from the type B Eulerian idempotents, not fitted to H*Z_n, so the cohomology identification is not a self-fulfilling fit. Theorem 1.2 uses the internal Pairing Lemma 5.2 to construct a bijection φ from the type A nbc-basis T to V∩∏Q, then proves gr(Y_n)^{Z_2} ≅ X_n = H*X_n inside the paper; the identification with higher Lie characters Lie_λ uses the classical Reutenauer/Garsia–Reutenauer isomorphism (2.18) Lie_λ ≅ (kS_n)E^{S_n}_λ, which is independent of the paper's conclusion. The cited results [15], [35], [42] are published theorems with independent proofs rather than assertions whose content is equivalent to the target. No fitted parameters, post-hoc exclusions, or definitions in terms of the target occur. One genuine limitation, noted by a close reading of the proof, is that Theorem 1.1(i),(iii) is stated with char(k)>2 but its proof invokes Theorem 2.5, which requires |W| ∈ k^×; for n≥3 and char p with 2<p≤n this is a gap. This is a correctness/robustness concern, not a circularity, because the proof does not assume the conclusion of Theorem 1.1 to prove it. Overall circularity score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim relies on published theorems for Varchenko-Gelfand rings, Eulerian idempotents, higher Lie characters, transfer, and semisimplicity, plus the explicitly stated field characteristic hypotheses. No unstated ad hoc assumptions or new entities are introduced.

assumptions (6)
  • standard math Moseley's identification VG(A) ≅ H^*(X_{R^3}) as graded k-algebras and k W-modules (Theorem 2.5).
    Invoked to express H^*X_n and H^*Y_n as Varchenko-Gelfand rings; a cited published theorem.
  • standard math The isomorphisms H^{2k}X_n ≅ (kS_n)E^{S_n}_{n-1-k} and H^{2k}Y_n ≅ (kS±_n)E^{S±_n}_{n-k} (Theorem 2.7, combining work of Hanlon, Sundaram-Welker, Brauner, and BBHT).
    The starting point for extending the known type A/B correspondence to peak idempotents.
  • standard math The finite-group transfer isomorphism H^*(Y/G) ≅ (H^*Y)^G when G acts freely and |G| ∈ k^× (Hatcher Prop 3G.1).
    Used in the proof of Theorem 1.1(i)-(ii) to identify H^*Z_n with the Z_2^n-fixed subalgebra of H^*Y_n.
  • standard math The identification Lie_lambda ≅ (kS_n)E^{S_n}_lambda ≅ VG(A_{S_n})_lambda (Reutenauer/Garsia, equation (2.18)).
    Converts the representation sums in Theorem 1.2 into higher Lie characters.
  • domain assumption Semisimplicity of kG when |G| ∈ k^×, permitting passage from a filtered kG-module to its associated graded (Section 2.6).
    The hypothesis char(k)>n is used pervasively in Propositions 6.5, 6.13, and Theorem 1.2 to replace filtered modules by direct sums of filtration quotients.
  • standard math The dimension of P_n/rad(P_n) equals the number of almost odd partitions of n (Aguiar-Nyman-Orellana [3, Cor. 4.3]).
    Used in Corollary 8.2 to prove that the finer peak idempotents E^{P_n}_mu are primitive.

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Pith. "Pith review of Configuration spaces and peak representations." pith.science (2026). https://pith.science/paper/ZZYNJBLH

@misc{pith2026250809898,
  author       = {Pith},
  title        = {Pith review of: Configuration spaces and peak representations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZYNJBLH}},
  note         = {Machine review of arXiv:2508.09898}
}
read the original abstract

Within the group algebras of the symmetric and hyperoctahedral groups, one has their descent algebras and families of Eulerian idempotents. These idempotents are known to generate group representations with topological interpretations, as the cohomology of configuration spaces of types A and B. We provide an analogous cohomological interpretation for the representations generated by idempotents in the peak algebra, called the peak representations. We describe the peak representations as sums of Thrall's higher Lie characters, give Hilbert series and branching rule recursions for them, and discuss a connection to Jordan brackets.

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Works this paper leans on

56 extracted references · 54 canonical work pages

  1. [1]

    Aguiar, N

    M. Aguiar, N. Bergeron, and K. Nyman. The peak algebra and the descent algebras of types B and D. Transactions of the American Mathematical Society, 356(7):2781–2824, 2004

    work page 2004

  2. [2]

    Aguiar and S

    M. Aguiar and S. Mahajan. Topics in hyperplane arrangements , volume 226. American Mathematical Soc., 2017

    work page 2017

  3. [3]

    Aguiar, K

    M. Aguiar, K. Nyman, and R. Orellana. New results on the peak algebra. J. Algebraic Combin. , 23(2):149–188, 2006

    work page 2006

  4. [4]

    Arnol’d, Vladimir

    I. Arnol’d, Vladimir. The cohomology ring of the group of dyed braids. Mat. Zametki , 5:227–231, 1969

    work page 1969

  5. [5]

    D. J. Benson. Representations and cohomology: Volume 1, basic representation theory of finite groups and associative algebras, volume 1. Cambridge university press, 1998

    work page 1998

  6. [6]

    Bergeron and N

    F. Bergeron and N. Bergeron. Orthogonal idempotents in the descent algebra of Bn and applications. J. Pure Appl. Algebra, 79(2):109–129, 1992

    work page 1992

  7. [7]

    Bergeron, N

    F. Bergeron, N. Bergeron, R. B. Howlett, and D. E. Taylor. A decomposition of the descent algebra of a finite Coxeter group. Journal of Algebraic Combinatorics , 1(1):23–44, 1992

    work page 1992

  8. [8]

    Bergeron and C

    N. Bergeron and C. Hohlweg. Coloured peak algebras and hopf algebras. Journal of Algebraic Combinatorics , 24(3):299– 330, 2006

    work page 2006

Show all 56 references
  1. [9]

    Bergeron, S

    N. Bergeron, S. Mykytiuk, F. Sottile, and S. van Willigenburg. Noncommutative pieri operators on posets. Journal of Combinatorial Theory, Series A , 91(1-2):84–110, 2000

    2000

  2. [10]

    Bidigare, P

    P. Bidigare, P. Hanlon, and D. Rockmore. A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements. Duke Math. J. , 99(1):135–174, 1999

    1999

  3. [11]

    T. P. Bidigare. Hyperplane arrangement face algebras and their associated Markov chains . ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)–University of Michigan

    1997

  4. [12]

    L. J. Billera, S. K. Hsiao, and S. van Willigenburg. Peak quasisymmetric functions and eulerian enumeration. Advances in Mathematics, 176(2):248–276, 2003

    2003

  5. [13]

    Billey and M

    S. Billey and M. Haiman. Schubert polynomials for the classical groups. Journal of the American Mathematical Society , 8(2):443–482, 1995

    1995

  6. [14]

    Bj¨ orner and G

    A. Bj¨ orner and G. M. Ziegler. Broken circuit complexes: factorizations and generalizations. J. Combin. Theory Ser. B , 51(1):96–126, 1991

    1991

  7. [15]

    S. Brauner. Eulerian representations for real reflection groups. Journal of the London Mathematical Society , 105(1):412– 444, 2022

    2022

  8. [16]

    Brauner, P

    S. Brauner, P. Commins, and V. Reiner. Invariant theory for the free left-regular band and a q-analogue. Pacific Journal of Mathematics , 322(2):251–280, 2023. CONFIGURATION SPACES AND PEAK REPRESENTATIONS 38

    2023

  9. [17]

    Calderbank, P

    A. Calderbank, P. Hanlon, and S. Sundaram. Representations of the symmetric group in deformations of the free Lie algebra. Transactions of the American mathematical Society , 341(1):315–333, 1994

    1994

  10. [18]

    F. R. Cohen, T. J. Lada, and P. J. May. The homology of iterated loop spaces , volume 533. Springer, 2007

    2007

  11. [19]

    P. Commins. Invariant theory for the face algebra of the braid arrangement. to appear in Algebraic Combinatorics , 2025

    2025

  12. [20]

    Cordovil

    R. Cordovil. A commutative algebra for oriented matroids. volume 27, pages 73–84. 2002. Geometric combinatorics (San Francisco, CA/Davis, CA, 2000)

    2002

  13. [21]

    D. A. Cox, J. Little, and D. O’Shea. Ideals, varieties, and algorithms . Undergraduate Texts in Mathematics. Springer, Cham, fourth edition, 2015. An introduction to computational algebraic geometry and commutative algebra

    2015

  14. [22]

    de Longueville and C

    M. de Longueville and C. A. Schultz. The cohomology rings of complements of subspace arrangements. Math. Ann. , 319(4):625–646, 2001

    2001

  15. [23]

    Dorpalen-Barry

    G. Dorpalen-Barry. The Varchenko-Gel’fand ring of a cone. J. Algebra, 617:500–521, 2023

    2023

  16. [24]

    A. M. Garsia. Combinatorics of the free Lie algebra and the symmetric group. In Analysis, et cetera , pages 309–382. Academic Press, Boston, MA, 1990

    1990

  17. [25]

    A. M. Garsia and C. Reutenauer. A decomposition of Solomon’s descent algebra. Advances in Mathematics, 77(2):189–262, 1989

    1989

  18. [26]

    Gerstenhaber and S

    M. Gerstenhaber and S. D. Schack. A Hodge-type decomposition for commutative algebra cohomology. J. Pure Appl. Algebra, 48(3):229–247, 1987

    1987

  19. [27]

    P. Hanlon. The action of Sn on the components of the Hodge decomposition of Hochschild homology. Michigan Math. J. , 37(1):105–124, 1990

    1990

  20. [28]

    A. Hatcher. Algebraic topology. Cambridge University Press, Cambridge, 2002

    2002

  21. [29]

    Hersh and V

    P. Hersh and V. Reiner. Representation stability for cohomology of configuration spaces in Rd. Int. Math. Res. Not. IMRN , (5):1433–1486, 2017. With an appendix written jointly with Steven Sam

    2017

  22. [30]

    S. K. Hsiao. Structure of the peak hopf algebra of quasi-symmetric functions. preprint, 2002

    2002

  23. [31]

    Kashuba and O

    I. Kashuba and O. Mathieu. On the free Jordan algebras. Adv. Math., 383:Paper No. 107690, 35, 2021

    2021

  24. [32]

    Y. Li. Representation theory of 0-hecke–clifford algebras. Journal of Algebra , 453:189–220, 2016

    2016

  25. [33]

    J.-L. Loday. Op´ erations sur l’homologie cyclique des alg` ebres commutatives. Inventiones mathematicae , 96(1):205–230, 1989

    1989

  26. [34]

    I. G. Macdonald. Symmetric functions and Hall polynomials . Oxford Classic Texts in the Physical Sciences. The Clarendon Press, Oxford University Press, New York, second edition, 2015. With contribution by A. V. Zelevinsky and a foreword by Richard Stanley, Reprint of the 2008...

    2015

  27. [35]

    D. Moseley. Equivariant cohomology and the Varchenko–Gelfand filtration. Journal of Algebra , 472:95–114, 2017

    2017

  28. [36]

    Novelli, F

    J.-C. Novelli, F. Saliola, and J.-Y. Thibon. Representation theory of the higher-order peak algebras. Journal of Algebraic Combinatorics, 32:465–495, 2010

    2010

  29. [37]

    K. L. Nyman. The peak algebra of the symmetric group. Journal of Algebraic Combinatorics , 17:309–322, 2003

    2003

  30. [38]

    Orlik and H

    P. Orlik and H. Terao. Arrangements of hyperplanes , volume 300 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1992

    1992

  31. [39]

    I. Peeva. Hyperplane arrangements and linear strands in resolutions. Trans. Amer. Math. Soc. , 355(2):609–618, 2003

    2003

  32. [40]

    Reutenauer

    C. Reutenauer. Theorem of Poincar´ e-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers. In Combinatoire ´ enum´ erative, pages 267–284. Springer, 1986

    1986

  33. [41]

    Reutenauer

    C. Reutenauer. Theorem of Poincar´ e-Birkhoff-Witt, logarithm and symmetric group representations of degrees equal to Stirling numbers. In Combinatoire ´ enum´ erative (Montreal, Que., 1985/Quebec, Que., 1985), volume 1234 of Lecture Notes in Math. , pages 267–284. Springer, B...

    1985

  34. [42]

    Reutenauer

    C. Reutenauer. Free Lie algebras. In Handbook of algebra, Vol. 3 , volume 3 of Handb. Algebr. , pages 887–903. Elsevier/North-Holland, Amsterdam, 2003

    2003

  35. [43]

    D. P. Robbins. Jordan elements in a free associative algebra, I. Journal of Algebra , 19(3):354–378, 1971

    1971

  36. [44]

    Schocker

    M. Schocker. Multiplicities of higher Lie characters. J. Aust. Math. Soc. , 75(1):9–21, 2003

    2003

  37. [45]

    Schocker

    M. Schocker. The peak algebra of the symmetric group revisited. Adv. Math., 192(2):259–309, 2005

    2005

  38. [46]

    L. Solomon. A Mackey formula in the group ring of a Coxeter group. Journal of Algebra , 41(2):255–264, 1976

    1976

  39. [47]

    R. P. Stanley. Enumerative combinatorics. Vol. 2 , volume 62 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1999. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin

    1999

  40. [48]

    R. P. Stanley. An introduction to hyperplane arrangements. In Geometric combinatorics , volume 13 of IAS/Park City Math. Ser. , pages 389–496. Amer. Math. Soc., Providence, RI, 2007

    2007

  41. [49]

    Stembridge

    J. Stembridge. Enriched P-partitions. Transactions of the American Mathematical Society , 349(2):763–788, 1997

    1997

  42. [50]

    Sundaram

    S. Sundaram. On a curious variant of the Sn-module Lie n. Algebr. Comb., 3(4):985–1009, 2020

    2020

  43. [51]

    Sundaram and V

    S. Sundaram and V. Welker. Group actions on arrangements of linear subspaces and applications to configuration spaces. Transactions of the American Mathematical Society , 349(4):1389–1420, 1997

    1997

  44. [52]

    R. M. Thrall. On symmetrized Kronecker powers and the structure of the free Lie ring. Amer. J. Math. , 64:371–388, 1942

    1942

  45. [53]

    Touchard

    J. Touchard. Sur les cycles des substitutions. Acta Math., 70(1):243–297, 1939

    1939

  46. [54]

    Uyemura-Reyes

    J.-C. Uyemura-Reyes. Random Walk, Semi-Direct Products, and Card Shuffling . PhD thesis, Stanford University, 2002

    2002

  47. [55]

    A. N. Varchenko and I. M. Gel’fand. Heaviside functions of a configuration of hyperplanes. Funktsional Anal. i Prilozhen , 21(4):1–18, 1987

    1987

  48. [56]

    M. A. Xicot´ encatl Merino. Orbit configuration spaces, infinitesimal braid relations in homology and equivariant loop spaces. PhD Thesis–University of Rochester. , 1997. CONFIGURATION SPACES AND PEAK REPRESENTATIONS 39 Appendix A. Symmetric function proofs of Theorem 9.4 and ...

    1997

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