View source for Hopf algebra of permutations

In algebra, the '''Malvenuto–Poirier–Reutenauer Hopf algebra of permutations''' or '''MPR Hopf algebra'''  is a [[Hopf algebra]] with a basis of all elements of all the finite symmetric groups ''S''<sub>''n''</sub>, and is a non-commutative analogue of the Hopf algebra of symmetric functions. It is both [[Free algebra|free]] as an [[Algebra (ring theory)|algebra]] and graded-cofree as a graded coalgebra, so is in some sense as far as possible from being either commutative or cocommutative. It was introduced by {{harvtxt|Malvenuto|Reutenauer|1995}} and studied by {{harvtxt|Poirier|Reutenauer|1995}}.
==Definition==

The underlying [[Free abelian group|free abelian group]] of the MPR algebra has a basis consisting of the disjoint union of the symmetric groups ''S''<sub>''n''</sub> for ''n'' = 0, 1, 2, .... , which can be thought of as permutations.

The identity 1 is the empty permutation, and the counit takes the empty permutation to 1 and the others to 0.

The product of two permutations (''a''<sub>1</sub>,...,''a''<sub>''m''</sub>) and  (''b''<sub>1</sub>,...,''b''<sub>''n''</sub>) in MPR is
given by the shuffle product  (''a''<sub>1</sub>,...,''a''<sub>''m''</sub>) ''ш'' (''m''&nbsp;+&nbsp;''b''<sub>1</sub>,...,''m''&nbsp;+&nbsp;''b''<sub>''n''</sub>).

The coproduct of a permutation ''a'' on ''m'' points is given by Σ<sub>''a''=''b''*''c''</sub>&nbsp;st(''b'')&nbsp;⊗&nbsp;st(''c''), where the sum is over the ''m''&nbsp;+&nbsp;1 ways to write ''a'' (considered as a sequence of ''m'' integers) as a concatenation of two sequences ''b'' and ''c'', and st(''b'') is the standardization of ''b'', where the elements of the sequence ''b'' are reduced to be a set of the form {1,&nbsp;2,&nbsp;...,&nbsp;''n''} while preserving their order.

The antipode has infinite order.

==Relation to other algebras==

The Hopf algebra of permutations relates the rings of [[Symmetric function|symmetric function]]s, [[Quasisymmetric function|quasisymmetric function]]s, and [[Noncommutative symmetric function|noncommutative symmetric function]]s, (denoted Sym, QSym, and NSym respectively), as depicted the following commutative diagram.  The duality between QSym and NSym is shown in the main diagonal of this diagram.

[[Image:QSymDiagram.png|300px|(Relationship between QSym and nearby neighbors)]]

==References==

*{{citation | mr=2724822 | zbl=1211.16023
|last1=Hazewinkel|first1= Michiel|last2= Gubareni|first2= Nadiya|last3= Kirichenko|first3= V. V.
|title=Algebras, rings and modules. Lie algebras and Hopf algebras|series= Mathematical Surveys and Monographs|volume= 168|publisher= American Mathematical Society|place= Providence, RI|year= 2010|isbn= 978-0-8218-5262-0 }}
*{{citation|mr=1358493 
|title=Duality between quasi-symmetric functions and the Solomon descent algebra
|journal=J. Algebra |volume=177 |year=1995|issue= 3|pages= 967–982|doi=10.1006/jabr.1995.1336|doi-access=free}}
*{{citation|mr=1334836 
|last1=Poirier|first1= Stéphane|last2= Reutenauer|first2= Christophe
|title=Algèbres de Hopf de tableaux
|journal=Ann. Sci. Math. Québec|volume= 19 |year=1995|issue= 1|pages= 79–90}}

[[Category:Hopf algebras]]

{{Sourceattribution|Hopf algebra of permutations}}