A pair of dual Hopf algebras on permutations

Mingze Zhao; Huilan Li; Mingze Zhao; Huilan Li

doi:10.3934/math.2021302

AIMS Mathematics

2021, Volume 6, Issue 5: 5106-5123. doi: 10.3934/math.2021302

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Research article

A pair of dual Hopf algebras on permutations

Mingze Zhao ,
Huilan Li ^,

School of Mathematics and Statistics, Shandong Normal University, Jinan, 250358, China

Received: 24 November 2020 Accepted: 18 February 2021 Published: 09 March 2021
MSC : 05A05, 08C20, 16T05

Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra.
- dual Hopf algebra,
- global descent,
- concatenation product,
- draw coproduct,
- antipode
Citation: Mingze Zhao, Huilan Li. A pair of dual Hopf algebras on permutations[J]. AIMS Mathematics, 2021, 6(5): 5106-5123. doi: 10.3934/math.2021302

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Abstract

Hopf algebras are important objects in algebraic combinatorics since they have strong stability. In particular, its dual space is an important tool to study the properties of the original Hopf algebra. Based on the classical shuffle Hopf algebra structure, we have proved that the shuffle product and deconcatenation coproduct on the standard factorizations of permutations define a graded shuffle Hopf algebra on permutations. In this paper, we figure out a new product and a new coproduct on permutations to get the duality of this graded shuffle Hopf algebra.

References

[1]	H. Hopf, Über die topologie der Gruppen-Mannigfaltigkeiten und ihrer verallgemeinerungen, In: Selecta Heinz Hopf, Berlin: Springer, 1941. Available from: https://doi.org/10.1007/978-3-662-25046-4_9.
[2]	J. W. Milnor, J. C. Moore, On the structure of Hopf algebras, Ann. Math., 81 (1965), 211–264. doi: 10.2307/1970615
[3]	M. E. Sweedler, Hopf Algebras, New York: W. A. Benjamin, Inc., 1969.
[4]	V. A. Artamonov, The structure of Hopf algebras, J. Math. Sci., 71 (1994), 2289–2328. doi: 10.1007/BF02111020
[5]	I. Kaplansky, Bialgebras, Department of Mathematics University of Chicago Chicago ILL, 2 (1975), 247–251.
[6]	N. van den Hijligenberg, R. Martini, Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra, J. Math. Phys., 37 (1996), 524. Available from: https://doi.org/10.1063/1.531407.
[7]	C. Reutenauer, Free Lie algebras, In: Handbook of Algebra, Amsterdam: Elsevier, 2003. Available from: https://doi.org/10.1016/S1570-7954(03)80075-X.
[8]	L. A. Duffaut Espinosa, K. Ebrahimi-Fard, W. S. Gray, A combinatorial hopf algebra for nonlinear output feedback control systems, J. Algebra, 453 (2016), 609–643. doi: 10.1016/j.jalgebra.2016.01.004
[9]	J. Høyrup, What is "geometric algebra", and what has it been in historiography?$^1$, AIMS Math., 2 (2017), 128–160. doi: 10.3934/Math.2017.1.128
[10]	J. Cheng, Y. Wang, R. B. Zhang, Degenerate quantum general linear groups, 2018. Available from: https://arXiv.org/abs/1805.07191.
[11]	T. Cheng, H. L. Huang, Y. P. Yang, Generalized Clifford algebras as algebras in suitable symmetric linear Gr-categories, SIGMA, 12 (2016), 1–11.
[12]	J. Tian, B. L. Li, Coalgebraic structure of genetic inheritance, Math. Biosci. Eng., 1 (2004), 243–266. doi: 10.3934/mbe.2004.1.243
[13]	M. Van den Bergh, A duality theorem for Hopf algebras, Methods Ring Theory, 129 (1984), 517–522.
[14]	R. J. Blattner, S. Montgomery, A duality theorem for Hopf module algebras, J. Algebra, 95 (1985), 153–172. doi: 10.1016/0021-8693(85)90099-7
[15]	S. A. Joni, G. C. Rota, Coalgebras and bialgebras in combinatorics, Stud. Appl. Math., 61 (1979), 93–139. doi: 10.1002/sapm197961293
[16]	M. Aguiar, N. Bergeron, F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math., 142 (2006), 1–30. doi: 10.1112/S0010437X0500165X
[17]	N. Bergeron, H. Li, Algebraic structures on Grothendieck groups of a tower of algebras, J. Algebra, 321 (2009), 2068–2084. doi: 10.1016/j.jalgebra.2008.12.005
[18]	N. Bergeron, H. Li, T. Lam, Combinatorial Hopf algebras and towers of algebras-dimension, quantization and functoriality, Algebr. Represent. Theory, 15 (2012), 675–696. doi: 10.1007/s10468-010-9258-y
[19]	H. Li, J. Morse, P. Shields, Structure constants for $K$-theory of Grassmannians, revisited, J. Comb. Theory, Ser. A, 144 (2016), 306–325. doi: 10.1016/j.jcta.2016.06.016
[20]	H. Li, T. MacHenry, A. Conci, Rational convolution roots of isobaric polynomials, Rocky Mt. J. Math., 4 (2017), 1259–1275.
[21]	L. Cao, Z. Chen, X. Duan, H. Li, Permanents of doubly substochastic matrices, Linear Multilinear Algebra, 68 (2020), 594–605. Available from: https://doi.org/10.1080/03081087.2018.1513448.
[22]	L. Cao, Z. Chen, Partitions of the polytope of doubly substochastic matrices, Linear Algebra Appl., 563 (2019), 98–122. doi: 10.1016/j.laa.2018.10.024
[23]	L. Cao, Z. Chen, X. Duan, S. Koyuncu, H. Li, Diagonal sums of doubly substochastic matrices, Electron. J. Linear Algebra, 35 (2017), 42–52.
[24]	L. Cao, A minimal completion of (0, 1)-matrices without total support, Linear Multilinear Algebra, 67 (2018), 1522–1530. Available from: https://doi.org/10.1080/03081087.2018.1460793.
[25]	R. Ree, Lie elements and an algebra associated with shuffles, Ann. Math., 68 (1958), 210–220. doi: 10.2307/1970243
[26]	K. T. Chen, An algebraic dualization of fundamental groups, Bull. Am. Math. Soc., 75 (1969), 1020–1024. doi: 10.1090/S0002-9904-1969-12345-1
[27]	M. Lothaire, Combinatorics on Words, UK: Cambridge University Press, 1997.
[28]	C. Malvenuto, C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177 (1995), 967–982. doi: 10.1006/jabr.1995.1336
[29]	A. M. Garsia, J. Remmel, Shuffles of permutations and the Kronecker product, Graphs Combinatorics, 1 (1985), 217–263. doi: 10.1007/BF02582950
[30]	M. Aguiar, F. Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, Adv. Math., 191 (2005), 225–275. doi: 10.1016/j.aim.2004.03.007
[31]	N. Bergeron, C. Ceballos, V. Pilaud, Hopf dreams and diagonal harmonics, 2018. Available from: https://arXiv.org/abs/1807.03044.
[32]	M. Zhao, H. Li, A Hopf algebra structure on permutations, J. Shangdong Norm. Univ. (Natual Sci.), 35 (2020), 417–422.
[33]	C. Schensted, Longest increasing and decreasing subsequences, Can. J. Math., 13 (1961), 179–191. doi: 10.4153/CJM-1961-015-3
[34]	Y. Vargas, Hopf algebra of permutation pattern functions, 26th International Conference on Formal Power Series and Algebraic Combinatorics, Chicago, United States, (2014), 839–850. Available from: https://hal.inria.fr/hal-01207565.
[35]	S. Fomin, Duality of graded graphs, J. Algebraic Combinatorics, 3 (1994), 357–404. Available from: https://doi.org/10.1023/A:1022412010826.
[36]	J. C. Aval, N. Bergeron, J. Machacek, New invariants for permutations, orders and graphs, Adv. Appl. Math., 121 (2020), 102080. doi: 10.1016/j.aam.2020.102080

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