ALGEBRAIC COMBINATORICS

no. 5
A natural idempotent in the descent algebra of a finite Coxeter group
Renteln, Paul1
1 California State University Department of Physics San Bernardino, CA 92407 USA
Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1177-1188.

We construct a natural idempotent in the descent algebra of a finite Coxeter group. The proof is uniform (independent of the classification). This leads to a simple determination of the spectrum of a natural matrix related to descents. Other applications are discussed.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.310
Classification: 05A05, 05C25, 05C50, 05E16
Keywords: Coxeter group, reflection representation, permutation representation, descents, descent algebra, idempotents, central limit theorems
Renteln, Paul 1

1 California State University Department of Physics San Bernardino, CA 92407 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2023__6_5_1177_0,
     author = {Renteln, Paul},
     title = {A natural idempotent in the descent algebra of a finite {Coxeter} group},
     journal = {Algebraic Combinatorics},
     pages = {1177--1188},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {5},
     year = {2023},
     doi = {10.5802/alco.310},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.310/}
}
TY  - JOUR
AU  - Renteln, Paul
TI  - A natural idempotent in the descent algebra of a finite Coxeter group
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 1177
EP  - 1188
VL  - 6
IS  - 5
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.310/
DO  - 10.5802/alco.310
LA  - en
ID  - ALCO_2023__6_5_1177_0
ER  - 
%0 Journal Article
%A Renteln, Paul
%T A natural idempotent in the descent algebra of a finite Coxeter group
%J Algebraic Combinatorics
%D 2023
%P 1177-1188
%V 6
%N 5
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.310/
%R 10.5802/alco.310
%G en
%F ALCO_2023__6_5_1177_0
Renteln, Paul. A natural idempotent in the descent algebra of a finite Coxeter group. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1177-1188. doi : 10.5802/alco.310. https://alco.centre-mersenne.org/articles/10.5802/alco.310/

[1] Adin, Ron M.; Roichman, Yuval The flag major index and group actions on polynomial rings, European J. Combin., Volume 22 (2001) no. 4, pp. 431-446 | DOI | MR | Zbl

[2] Atkinson, M. D. A new proof of a theorem of Solomon, Bull. London Math. Soc., Volume 18 (1986) no. 4, pp. 351-354 | DOI | MR | Zbl

[3] Atkinson, M. D. Solomon’s descent algebra revisited, Bull. London Math. Soc., Volume 24 (1992) no. 6, pp. 545-551 | DOI | MR | Zbl

[4] Bergeron, F.; Bergeron, N.; Howlett, R. B.; Taylor, D. E. A decomposition of the descent algebra of a finite Coxeter group, J. Algebraic Combin., Volume 1 (1992) no. 1, pp. 23-44 | DOI | MR | Zbl

[5] Bidigare, Pat; Hanlon, Phil; Rockmore, Dan A combinatorial description of the spectrum for the Tsetlin library and its generalization to hyperplane arrangements, Duke Math. J., Volume 99 (1999) no. 1, pp. 135-174 | DOI | MR | Zbl

[6] Bidigare, Thomas Patrick Hyperplane arrangement face algebras and their associated Markov chains, Ph. D. Thesis, University of Michigan (1997), vii+151 pages | MR

[7] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | MR

[8] Bóna, Miklós Combinatorics of permutations, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2012, xiv+458 pages | DOI | MR

[9] Bourbaki, Nicolas Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002, xii+300 pages (translated from the 1968 French original by Andrew Pressley) | DOI | MR

[10] Brown, Kenneth S. Semigroups, rings, and Markov chains, J. Theoret. Probab., Volume 13 (2000) no. 3, pp. 871-938 | DOI | MR | Zbl

[11] Brück, Benjamin; Röttger, Frank A central limit theorem for the two-sided descent statistic on Coxeter groups, Electron. J. Combin., Volume 29 (2022) no. 1, Paper no. 1.1, 25 pages | DOI | MR | Zbl

[12] Carlitz, L. q-Bernoulli and Eulerian numbers, Trans. Amer. Math. Soc., Volume 76 (1954), pp. 332-350 | DOI | MR | Zbl

[13] Chatterjee, Sourav; Diaconis, Persi A central limit theorem for a new statistic on permutations, Indian J. Pure Appl. Math., Volume 48 (2017) no. 4, pp. 561-573 | DOI | MR | Zbl

[14] Comtet, Louis Advanced combinatorics: the art of finite and infinite expansions, D. Reidel Publishing Co., Dordrecht, 1974, xi+343 pages | DOI | MR

[15] Féray, Valentin On the central limit theorem for the two-sided descent statistics in Coxeter groups, Electron. Commun. Probab., Volume 25 (2020), Paper no. 28, 6 pages | DOI | MR | Zbl

[16] Foata, Dominique On the Netto inversion number of a sequence, Proc. Amer. Math. Soc., Volume 19 (1968), pp. 236-240 | DOI | MR | Zbl

[17] Foata, Dominique; Schützenberger, Marcel-Paul Major index and inversion number of permutations, Math. Nachr., Volume 83 (1978), pp. 143-159 | DOI | MR | Zbl

[18] Garsia, A. M.; Gessel, I. Permutation statistics and partitions, Adv. in Math., Volume 31 (1979) no. 3, pp. 288-305 | DOI | MR | Zbl

[19] Garsia, A. M.; Reutenauer, C. A decomposition of Solomon’s descent algebra, Adv. Math., Volume 77 (1989) no. 2, pp. 189-262 | DOI | MR | Zbl

[20] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | DOI | MR

[21] Kahle, Thomas; Stump, Christian Counting inversions and descents of random elements in finite Coxeter groups, Math. Comp., Volume 89 (2020) no. 321, pp. 437-464 | DOI | MR | Zbl

[22] Kane, Richard Reflection groups and invariant theory, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 5, Springer-Verlag, New York, 2001, x+379 pages | DOI | MR

[23] Petersen, T. Kyle Eulerian numbers, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser/Springer, New York, 2015, xviii+456 pages | DOI | MR

[24] Randriamaro, Hery Diagonalization of matrices of statistics, Ann. Comb., Volume 17 (2013) no. 3, pp. 549-569 | DOI | MR | Zbl

[25] Randriamaro, Hery Spectral properties of descent algebra elements, J. Algebraic Combin., Volume 39 (2014) no. 1, pp. 127-139 | DOI | MR | Zbl

[26] Randriamaro, Hery Diagonalization of fix-Mahonian matrices, 2020 | arXiv

[27] Reiner, Victor; Saliola, Franco; Welker, Volkmar Spectra of symmetrized shuffling operators, Mem. Amer. Math. Soc., Volume 228 (2014) no. 1072, p. vi+109 | MR | Zbl

[28] Renteln, Paul The distance spectra of Cayley graphs of Coxeter groups, Discrete Math., Volume 311 (2011) no. 8-9, pp. 738-755 | DOI | MR | Zbl

[29] Röttger, Frank Asymptotics of a locally dependent statistic on finite reflection groups, Electron. J. Combin., Volume 27 (2020) no. 2, Paper no. 2.24, 11 pages | DOI | MR | Zbl

[30] Saliola, Franco V. The face semigroup algebra of a hyperplane arrangement, Ph. D. Thesis, Cornell University (2006), viii+109 pages | MR

[31] Saliola, Franco V. Hyperplane arrangements and descent algebras, 2006 (unpublished notes)

[32] Saliola, Franco V. On the quiver of the descent algebra, J. Algebra, Volume 320 (2008) no. 11, pp. 3866-3894 | DOI | MR | Zbl

[33] Solomon, Louis A Mackey formula in the group ring of a Coxeter group, J. Algebra, Volume 41 (1976) no. 2, pp. 255-264 | DOI | MR | Zbl

[34] Stanley, Richard P. Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR

[35] Steinberg, Robert Finite reflection groups, Trans. Amer. Math. Soc., Volume 91 (1959), pp. 493-504 | DOI | MR | Zbl

[36] Tsilevich, N. V. On the dual complexity and spectra of some combinatorial functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 462 (2017), pp. 112-121 (Teoriya Predstavleniĭ, Dinamicheskie Sistemy, Kombinatornye Metody. XXVIII) | DOI | MR

[37] van Willigenburg, Stephanie A proof of Solomon’s rule, J. Algebra, Volume 206 (1998) no. 2, pp. 693-698 | DOI | MR | Zbl

[38] Vershik, A. M.; Tsilevich, N. V. On the relationship between combinatorial functions and representations of a symmetric group, Funktsional. Anal. i Prilozhen., Volume 51 (2017) no. 1, pp. 28-39 | DOI | MR

Cited by Sources: