The Frobenius transform of a symmetric function

Lee, Mitchell

doi:10.5802/alco.365

Lee, Mitchell ¹

¹ Harvard University Dept. of mathematics 1 Oxford st. Cambridge MA 02138 (USA)

Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 931-958.

Abstract

We define an abelian group homomorphism $ℱ$ , which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $ℱ$ in the Schur basis are the restriction coefficients $r_{λ}^{μ} = dim {Hom}_{𝔖_{n}} (V_{μ}, 𝕊^{λ} ℂ^{n})$ , which are known to be nonnegative integers but have no known combinatorial interpretation. The Frobenius transform satisfies the identity $ℱ {f g} = ℱ {f} * ℱ {g}$ , where $*$ is the Kronecker product.

We prove for all symmetric functions $f$ that $ℱ {f} = ℱ_{Sur} \{f\} \cdot (1 + h_{1} + h_{2} + \dots)$ , where $ℱ_{Sur} \{f\}$ is a symmetric function with the same degree and leading term as $f$ . Then, we compute the matrix entries of $ℱ_{Sur}$ in the complete homogeneous, elementary, and power sum bases and of $ℱ_{Sur}^{- 1}$ in the complete homogeneous and elementary bases, giving combinatorial interpretations of the coefficients where possible. In particular, the matrix entries of $ℱ_{Sur}^{- 1}$ in the elementary basis count words with a constraint on their Lyndon factorization.

As an example application of our main results, we prove that $r_{λ}^{μ} = 0$ if $| λ \cap \hat{μ} | < 2 | \hat{μ} | - | λ |$ , where $\hat{μ}$ is the partition formed by removing the first part of $μ$ . We also prove that $r_{λ}^{μ} = 0$ if the Young diagram of $μ$ contains a square of side length greater than $2^{λ_{1} - 1}$ , and this inequality is tight.

Received: 2023-09-06
Accepted: 2024-02-19
Published online: 2024-09-03

DOI: 10.5802/alco.365

Classification: 05E05
Keywords: symmetric functions, representation theory of categories, plethysm, Kronecker product

Author's affiliations:

Lee, Mitchell ¹

¹ Harvard University Dept. of mathematics 1 Oxford st. Cambridge MA 02138 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2024__7_4_931_0,
     author = {Lee, Mitchell},
     title = {The {Frobenius} transform of a symmetric function},
     journal = {Algebraic Combinatorics},
     pages = {931--958},
     publisher = {The Combinatorics Consortium},
     volume = {7},
     number = {4},
     year = {2024},
     doi = {10.5802/alco.365},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.365/}
}

TY  - JOUR
AU  - Lee, Mitchell
TI  - The Frobenius transform of a symmetric function
JO  - Algebraic Combinatorics
PY  - 2024
SP  - 931
EP  - 958
VL  - 7
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.365/
DO  - 10.5802/alco.365
LA  - en
ID  - ALCO_2024__7_4_931_0
ER  -

%0 Journal Article
%A Lee, Mitchell
%T The Frobenius transform of a symmetric function
%J Algebraic Combinatorics
%D 2024
%P 931-958
%V 7
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.365/
%R 10.5802/alco.365
%G en
%F ALCO_2024__7_4_931_0

Lee, Mitchell. The Frobenius transform of a symmetric function. Algebraic Combinatorics, Volume 7 (2024) no. 4, pp. 931-958. doi : 10.5802/alco.365. https://alco.centre-mersenne.org/articles/10.5802/alco.365/

References
Cited by

[1] Andrews, George E.; Eriksson, Kimmo Integer partitions, Cambridge University Press, Cambridge, 2004, x+141 pages | DOI | MR

[2] Assaf, Sami H.; Speyer, David E. Specht modules decompose as alternating sums of restrictions of Schur modules, Proc. Amer. Math. Soc., Volume 148 (2020) no. 3, pp. 1015-1029 | DOI | MR | Zbl

[3] Cadogan, C. C. The Möbius function and connected graphs, J. Combinatorial Theory Ser. B, Volume 11 (1971), pp. 193-200 | DOI | MR | Zbl

[4] Chen, K.-T.; Fox, R. H.; Lyndon, R. C. Free differential calculus. IV. The quotient groups of the lower central series, Ann. of Math. (2), Volume 68 (1958), pp. 81-95 | DOI | MR | Zbl

[5] Fulton, William Young tableaux: with applications to representation theory and geometry, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages | MR

[6] Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren Concrete mathematics: a foundation for computer science, Addison-Wesley Publishing Company, Reading, MA, 1994, xiv+657 pages | MR

[7] Joyal, André Foncteurs analytiques et espèces de structures, Combinatoire énumérative (Montreal, Que., 1985/Quebec, Que., 1985) (Lecture Notes in Math.), Volume 1234, Springer, Berlin, 1986, pp. 126-159 | DOI | MR | Zbl

[8] Littlewood, D. E. Group Characters and the Structure of Groups, Proc. London Math. Soc. (2), Volume 39 (1935) no. 2, pp. 150-199 | DOI | MR | Zbl

[9] Loehr, Nicholas A.; Remmel, Jeffrey B. A computational and combinatorial exposé of plethystic calculus, J. Algebraic Combin., Volume 33 (2011) no. 2, pp. 163-198 | DOI | MR | Zbl

[10] Lothaire, M. Combinatorics on words, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997, xviii+238 pages | DOI | MR

[11] Lyndon, R. C. On Burnside’s problem. II, Trans. Amer. Math. Soc., Volume 78 (1955), pp. 329-332 | DOI | MR | Zbl

[12] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Classic Texts in the Physical Sciences, The Clarendon Press, Oxford University Press, New York, 2015, xii+475 pages | MR

[13] Méndez, M. Tensor species and symmetric functions, Proc. Nat. Acad. Sci. U.S.A., Volume 88 (1991) no. 21, pp. 9892-9894 | DOI | MR | Zbl

[14] Narayanan, Sridhar P.; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha Character polynomials and the restriction problem, Algebr. Comb., Volume 4 (2021) no. 4, pp. 703-722 | DOI | Numdam | MR | Zbl

[15] Narayanan, Sridhar P.; Paul, Digjoy; Prasad, Amritanshu; Srivastava, Shraddha Polynomial induction and the restriction problem, Indian J. Pure Appl. Math., Volume 52 (2021) no. 3, pp. 643-651 | DOI | MR | Zbl

[16] Orellana, Rosa; Zabrocki, Mike A combinatorial model for the decomposition of multivariate polynomial rings as $S_{n}$ -modules, Electron. J. Combin., Volume 27 (2020) no. 3, Paper no. 3.24, 18 pages | DOI | MR | Zbl

[17] Orellana, Rosa; Zabrocki, Mike The Hopf structure of symmetric group characters as symmetric functions, Algebr. Comb., Volume 4 (2021) no. 3, pp. 551-574 | DOI | Numdam | MR | Zbl

[18] Orellana, Rosa; Zabrocki, Mike Symmetric group characters as symmetric functions, Adv. Math., Volume 390 (2021), Paper no. 107943, 34 pages | DOI | MR | Zbl

[19] Reutenauer, Christophe Free Lie algebras, Handbook of algebra, Vol. 3 (Handb. Algebr.), Volume 3, Elsevier/North-Holland, Amsterdam, 2003, pp. 887-903 | DOI | MR | Zbl

[20] Ryba, Christopher Resolving irreducible $ℂ S_{n}$ -modules by modules restricted from $G L_{n} (ℂ)$ , Represent. Theory, Volume 24 (2020), pp. 229-234 | DOI | MR | Zbl

[21] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 208, Cambridge University Press, Cambridge, 2024, xvi+783 pages | MR

[22] Wiltshire-Gordon, John D. Representation Theory of Combinatorial Categories, Ph. D. Thesis, University of Michigan (2016) (113 pages)

[23] Zabrocki, Mike Vertex operators for standard bases of the symmetric functions, J. Algebraic Combin., Volume 13 (2001) no. 1, pp. 83-101 | DOI | MR | Zbl

Cited by Sources: