Primitive elements of the Hopf algebras of tableaux
Introduction
In 1995 the authors of the present paper introduced two dual Hopf algebra structures on permutations [11]. The products and coproducts of permutations originated from the concatenation Hopf algebra and shuffle Hopf algebra on , the module generated by words of positive integers, from Solomon’s descent algebra [15] and Gessel’s (internal) coalgebra [5] of quasi-symmetric functions. The two Hopf structures on , the module with –basis all the permutations in , for the symmetric group on , are self-dual (this result is somewhat implicit in the earlier work of the authors of the present article, but is explicit in [7] and [2]).
In the same year [13], carrying on these themes, Poirier and the second author proved that the two dual Hopf structures on are free associative algebras. By restriction on the plactic classes they obtained two dual structures of Hopf algebras on the –module with basis the set of all standard Young tableaux. The product and coproduct are described there in terms of Schützenberger’s “jeu de taquin” [9]. They also provided different morphisms between these structures and the descent algebras, symmetric functions and quasi-symmetric functions. In particular, the map sending a permutation into its left tableau in the Schensted correspondence is a Hopf morphism.
Loday and Ronco [10] characterized the product of two permutations by the use of the weak order of permutations: it is the sum of all permutations in some interval for this order. In 2005 [1], Aguiar and Sottile studied in further and thorough details the structure of the Hopf algebras of permutations, giving explicit formulas for its antipode, proving that it is a cofree algebra and determining its primitive elements. For the latter task, they introduced a new basis of , related to the basis of permutations via Möbius inversion in the poset of the Bruhat weak order of the symmetric groups. In [4], Duchamp, Hivert, Novelli and Thibon studied the Hopf algebra of permutations (denoted there ), and gave among others a faithful representation by noncommutative polynomials.
The Hopf algebra of tableaux was used by Jöllenbeck [7], and Blessenohl and Schocker [2], to define their noncommutative character theory of the symmetric group. Moreover, Muge Tasķin [16] used the order on tableaux, induced by the weak order of permutations, to characterize the product of two tableaux, in a way reminiscent of the result of Loday and Ronco.
The purpose of this Note is to find the primitive elements of , the Hopf algebra of tableaux with respect to the product and coproduct, following the approach of Aguiar and Sottile. A new basis for is obtained by Möbius inversion for the Tasķin order of tableaux. The nice feature of the proofs here is that we manage to avoid “jeu de taquin”, and use a simpler description through a shifted left and right concatenation product of tableaux.
Note that Tasķin studies in [16] four orders on tableaux, which appeared in earlier work of Melnikov [12]; she calls the weak order on tableaux the induced Duflo order. The four orders are mutually comparable, and the weak order is the smallest one. Moreover, Tasķin, with Karaali and Senturia, introduced recently a new order on Young tableaux [8]. We did not explore what happens if one performs an change of basis based on these orders (except the weak order). It could be that the primitive elements of behave nicely in this new basis, and that moreover the multiplicative constants in one of these basis are positive; but this is open.
Section snippets
Preliminaries on permutations
We denote by symmetric group on . We often represent permutations as words: is represented as the word . By abuse of notation, we identify and the corresponding word. A word in the sequel will always be on the alphabet of positive integers, also called letters. We denote by the number of letters of .
We denote by the right weak order of permutations. Recall that it is defined as the reflexive and transitive closure of the relation , , , for some
Hopf algebra of permutations
Denote by be the free -module with basis . We define on a product, denoted by (called destandardized concatenation), and a coproduct (called standardized unshuffling), denoted by , which turn it into a Hopf algebra (see [11]). If , , is the sum of all permutations in of the form (concatenation of and ), where are of respective lengths and , ; for example, . Moreover, for ,
Preliminaries on tableaux
For unreferenced results quoted here, see [14]. Denote by denotes the set of standard Young tableaux (we say simply tableaux) whose entries are . We denote by the pair of tableaux associated with by the Schensted correspondence.
Let be the set of all standard tableaux. The plactic equivalence on is the smallest equivalence relation generated by the Knuth relations , , , for and .
By Knuth’s theorem, one has
Main lemma
Recall the Taskin weak order on tableaux, denoted .
Lemma 5.1 Let and . Let , and let . Then for , , one has: if and only if and .
Proof of Lemma 5.1 1. We show first that . Let be such that . Let and . Then by Lemma 3.1. It follows that by Lemma 4.1, Lemma 4.2. 2. Suppose that and . Then by 1. and Lemma 4.2, we have . 3. Suppose now that . Let be such that .
Primitive elements in the Hopf algebra of tableaux
The free -module , based on the set of tableaux, becomes a structure of Hopf algebra, quotient of the Hopf algebra of Section 3, and whose product and coproduct are therefore also denoted by and . The quotient is obtained by identifying plactic equivalent permutations. In other words, consider the submodule spanned by the elements , ; then is an ideal and a co-ideal of , and the quotient is canonically isomorphic with . Moreover, the canonical bialgebra
Product formulas using
The product , both for permutations and tableaux, plays a role in product formulas in the dual Hopf algebras of and .
First, one has to consider also the product of permutations: let , ; then , where is obtained from by adding to each letter in .
Recall the shifted shuffle product of permutations, denoted : is the shuffle of and . This product is the dual product of the coproduct of . On has
Theorem 7.1 Let . Then Loday-Ronco [10] Theorem 4.1
In other words, is the
Acknowledgments
We thank Franco Saliola, who allowed us to include a counter-example arising from his computations. This work was partially supported by a grant of NSERC (Canada) for the second author, and was partially done during an invitation of him to the university La Sapienza in 2019.
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The collected papers of Schützenberger are available on the website of Jean Berstel.