Hopf algebra structure of symmetric and quasisymmetric functions in superspace

Luc Lapointe; Maria Elena Pinto; Susanna Fishel

arxiv: 1907.09975 · v1 · pith:M7GSSR7Enew · submitted 2019-07-23 · 🧮 math.CO

Hopf algebra structure of symmetric and quasisymmetric functions in superspace

Susanna Fishel , Luc Lapointe , Maria Elena Pinto This is my paper

Pith reviewed 2026-05-24 17:22 UTC · model grok-4.3

classification 🧮 math.CO

keywords symmetric functions in superspacequasisymmetric functionsHopf algebranoncommutative symmetric functionscocommutativeself-dualmonomial basisantipode

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The pith

The ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra, and the ring of quasisymmetric functions in superspace is a Hopf algebra with a multiplicative dual basis in noncommutative symmetric functions.

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

The paper shows that symmetric functions defined on superspace variables carry the structure of a cocommutative Hopf algebra that is self-dual. Formulas are provided for how the coproduct and antipode act on standard bases of this ring. The authors introduce quasisymmetric functions in superspace as a Hopf algebra and give explicit rules for multiplying and comultiplying its monomial basis elements. They prove that the dual Hopf algebra consists of noncommutative symmetric functions in superspace whose basis multiplies in a simple way dual to the monomial quasisymmetric basis.

Core claim

We show that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra. We provide formulas for the action of the coproduct and the antipode on various bases of that ring. We introduce the ring sQSym of quasisymmetric functions in superspace and show that it is a Hopf algebra. We give explicitly the product, coproduct and antipode on the basis of monomial quasisymmetric functions in superspace. We prove that the Hopf dual of sQSym, the ring sNSym of noncommutative symmetric functions in superspace, has a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

What carries the argument

The monomial bases of symmetric and quasisymmetric functions in superspace together with the explicitly defined coproduct and antipode maps that turn the rings into Hopf algebras.

Load-bearing premise

The chosen definitions of the superspace variables, the monomial bases, and the coproduct and antipode maps satisfy the Hopf algebra axioms without requiring extra relations or restrictions beyond those stated in the constructions.

What would settle it

A direct computation on a low-degree monomial element showing that the twice-applied coproduct fails to be coassociative or that the antipode fails to satisfy the required convolution identity with the unit and counit would disprove the Hopf algebra claim.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

This paper adds explicit Hopf structures on new superspace versions of symmetric and quasisymmetric functions, with direct formulas and duality.

read the letter

The core contribution is the introduction of sQSym and sNSym together with verified Hopf operations on their bases and a duality statement between them. The symmetric functions in superspace are shown to be cocommutative and self-dual, while sQSym carries a Hopf structure whose monomial basis is dual to a multiplicative basis in sNSym. Explicit coproduct and antipode formulas are supplied on the chosen bases, followed by direct checks of the axioms. That is the actual new material; the classical case is recovered when the superspace variables are set to zero. The work is straightforward algebraic combinatorics and stays within the standard direct-verification style for graded Hopf algebras. The definitions of the superspace monomials and the maps appear consistent with the abstract claims, and no circularity or hidden fitting is visible. A minor limitation is that the naturalness of the superspace grading and basis choices is not compared against other possible extensions, but the paper does not claim uniqueness. Readers already working on combinatorial Hopf algebras or supersymmetric extensions will find the explicit bases and duality useful for further calculations. The paper is coherent on its own terms and deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 3 minor

Summary. The manuscript shows that the ring of symmetric functions in superspace is a cocommutative and self-dual Hopf algebra, supplying explicit formulas for the coproduct and antipode on its bases. It introduces the ring sQSym of quasisymmetric functions in superspace, proves it is a Hopf algebra, and gives explicit product, coproduct, and antipode formulas on the monomial basis; it further shows that the dual ring sNSym admits a multiplicative basis dual to the monomial quasisymmetric functions in superspace.

Significance. The explicit constructions and verifications extend the classical Hopf structures on Sym and QSym to the superspace setting. The self-duality result for symmetric functions in superspace and the explicit duality between the monomial basis of sQSym and a multiplicative basis of sNSym are concrete strengths that support further combinatorial and algebraic work.

minor comments (3)

[§1] §1: the notation for the superspace variables (e.g., the distinction between even and odd generators) is introduced only briefly; a short dedicated paragraph would improve readability for readers unfamiliar with the superspace setting.
The paper states that the Hopf axioms are verified by direct computation on the chosen bases, but the verification steps for coassociativity of the coproduct on the monomial basis of sQSym are only sketched; expanding the key calculation in an appendix would strengthen the presentation.
Table 1 (basis comparisons) uses inconsistent column headings between the classical and superspace cases; aligning the notation would reduce reader confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments appear in the report, so there are no specific points requiring point-by-point rebuttal or revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper performs direct algebraic constructions of the superspace symmetric and quasisymmetric function rings, supplies explicit basis formulas for the product, coproduct and antipode, and verifies the Hopf algebra axioms (coassociativity, counit, antipode) by explicit calculation on those bases. No load-bearing step reduces to a fitted parameter, a self-citation chain, or a definition that presupposes the target result. The central claims are therefore self-contained against the stated definitions and do not rely on any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no explicit free parameters, ad-hoc axioms, or invented entities are visible. The constructions rest on the standard definition of superspace (commuting plus anticommuting variables) and the usual Hopf algebra axioms, which are treated as background.

axioms (1)

standard math Standard Hopf algebra axioms (coassociativity, counit, antipode) hold for the defined coproduct and antipode on the superspace rings.
Invoked implicitly when the abstract states that the rings 'are' Hopf algebras and supplies formulas for the operations.

pith-pipeline@v0.9.0 · 5642 in / 1441 out tokens · 22478 ms · 2026-05-24T17:22:28.256503+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

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O. Blondeau-Fournier, P . Desrosiers, L. Lapointe and P . Mathieu, Macdonald polynomials in superspace as eigenfunctions of c ommuting operators, Journal of Combinatorics 3, no. 3, 495–562 (2012)

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Blondeau-Fournier and P

O. Blondeau-Fournier and P . Mathieu, Schur superpolynomials: Combinatorial deﬁnition and Pier i Rule, SIGMA 11 (2015), 021, 23pp

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Desrosiers, L

P . Desrosiers, L. Lapointe and P . Mathieu, Classical symmetric functions in superspace , J. Alg. Combin. 24, 209–238 (2006)

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Desrosiers, L

P . Desrosiers, L. Lapointe and P . Mathieu, Orthogonality of Jack polynomials in superspace , Adv. Math. 212, 361–388 (2007)

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Desrosiers, L

P . Desrosiers, L. Lapointe and P . Mathieu, Evaluation and normalization of Jack superpolynomials , IMRN 23, 5267–5327 (2012)

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[7]

Ehrenborg, On posets and Hopf algebras , Adv

R. Ehrenborg, On posets and Hopf algebras , Adv. Math. 119, 1–25 (1996)

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Fishel, L

S. Fishel, L. Lapointe and M.E. Pinto, Fundamental quasisymmetric functions in superspace , in preparation

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Grinberg and V

D. Grinberg and V . Reiner, Hopf algebras in combinatorics, arXiv:1409.8356

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Haglund, M

J. Haglund, M. Haiman and N. Loehr, A combinatorial formula for Macdonald polynomials , J. Amer. Math. Soc. 18, 735–761 (2005)

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[13]

Jones and L

M. Jones and L. Lapointe, Pieri rules for the Schur functions in superspace , J. Combin. Theory Ser. A 148, 57–115 (2017)

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Luoto, S

K. Luoto, S. Mykytiuk and S. van Willigenburg, An introd uction to quasisymmetric Schur functions. Hopf algebras, q uasisymmetric functions, and Y oung composition tableaux, SpringerBriefs in Mathematics. Springer, New Y ork, 2013

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R. P . Stanley, Enumerative Combinatorics, V ol. 2, Cambridge Studies in Advanced Mathematics V ol. 62, Cambridge University Press, 1999. SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES , A RIZONA STATE UNIVERSITY , P.O. B OX 871804, T EMPE , AZ 85287-1804, USA E-mail address: sfishel1@asu.edu INSTITUTO DE MATEMÁTICA Y FÍSICA , U NIVERSIDAD DE TALCA , C A...

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[1] [1]

Blondeau-Fournier, P

O. Blondeau-Fournier, P . Desrosiers, L. Lapointe and P . Mathieu, Macdonald polynomials in superspace as eigenfunctions of c ommuting operators, Journal of Combinatorics 3, no. 3, 495–562 (2012)

work page 2012

[2] [2]

Blondeau-Fournier and P

O. Blondeau-Fournier and P . Mathieu, Schur superpolynomials: Combinatorial deﬁnition and Pier i Rule, SIGMA 11 (2015), 021, 23pp

work page 2015

[3] [3]

Doliwa, Hopf algebra structure of generalized quasi- symmetric functions in partially commutative variables, a rXiv:1603.03259

A. Doliwa, Hopf algebra structure of generalized quasi- symmetric functions in partially commutative variables, a rXiv:1603.03259

work page arXiv

[4] [4]

Desrosiers, L

P . Desrosiers, L. Lapointe and P . Mathieu, Classical symmetric functions in superspace , J. Alg. Combin. 24, 209–238 (2006)

work page 2006

[5] [5]

Desrosiers, L

P . Desrosiers, L. Lapointe and P . Mathieu, Orthogonality of Jack polynomials in superspace , Adv. Math. 212, 361–388 (2007)

work page 2007

[6] [6]

Desrosiers, L

P . Desrosiers, L. Lapointe and P . Mathieu, Evaluation and normalization of Jack superpolynomials , IMRN 23, 5267–5327 (2012)

work page 2012

[7] [7]

Ehrenborg, On posets and Hopf algebras , Adv

R. Ehrenborg, On posets and Hopf algebras , Adv. Math. 119, 1–25 (1996)

work page 1996

[8] [8]

Fishel, L

S. Fishel, L. Lapointe and M.E. Pinto, Fundamental quasisymmetric functions in superspace , in preparation

work page

[9] [9]

Gelfand, D

I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V . Retakh, J. -Y . Thibon,Noncommutative symmetric functions, Adv. Math. 112, 218–348 (1995)

work page 1995

[10] [10]

Gessel, Multipartite P -partitions and inner products of skew Schur functions , Contemp

I. Gessel, Multipartite P -partitions and inner products of skew Schur functions , Contemp. Math 34, 289–301 (1984)

work page 1984

[11] [11]

Grinberg and V

D. Grinberg and V . Reiner, Hopf algebras in combinatorics, arXiv:1409.8356

work page arXiv

[12] [12]

Haglund, M

J. Haglund, M. Haiman and N. Loehr, A combinatorial formula for Macdonald polynomials , J. Amer. Math. Soc. 18, 735–761 (2005)

work page 2005

[13] [13]

Jones and L

M. Jones and L. Lapointe, Pieri rules for the Schur functions in superspace , J. Combin. Theory Ser. A 148, 57–115 (2017)

work page 2017

[14] [14]

Luoto, S

K. Luoto, S. Mykytiuk and S. van Willigenburg, An introd uction to quasisymmetric Schur functions. Hopf algebras, q uasisymmetric functions, and Y oung composition tableaux, SpringerBriefs in Mathematics. Springer, New Y ork, 2013

work page 2013

[15] [15]

I. G. Macdonald, Symmetric functions and Hall polynomi als, 2nd ed., Clarendon Press, 1995

work page 1995

[16] [16]

R. P . Stanley, Enumerative Combinatorics, V ol. 2, Cambridge Studies in Advanced Mathematics V ol. 62, Cambridge University Press, 1999. SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES , A RIZONA STATE UNIVERSITY , P.O. B OX 871804, T EMPE , AZ 85287-1804, USA E-mail address: sfishel1@asu.edu INSTITUTO DE MATEMÁTICA Y FÍSICA , U NIVERSIDAD DE TALCA , C A...

work page 1999