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Hopf algebra of permutations
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<h2 id="definition">Definition</h2> <p>The underlying <a href="/facts/Free_abelian_group/oaSRGmpY">free abelian group</a> of the MPR algebra has a basis consisting of the disjoint union of the symmetric groups <i>S</i><i>n</i> for <i>n</i> = 0, 1, 2, .... , which can be thought of as permutations. </p><p>The identity 1 is the empty permutation, and the counit takes the empty permutation to 1 and the others to 0. </p><p>The product of two permutations (<i>a</i>1,...,<i>a</i><i>m</i>) and (<i>b</i>1,...,<i>b</i><i>n</i>) in MPR is given by the <a href="/facts/Shuffle_product/AgsGQqgJ">shuffle product</a> (<i>a</i>1,...,<i>a</i><i>m</i>) <i>ш</i> (<i>m</i> + <i>b</i>1,...,<i>m</i> + <i>b</i><i>n</i>). </p><p>The coproduct of a permutation <i>a</i> on <i>m</i> points is given by Σ<i>a</i>=<i>b</i>*<i>c</i> st(<i>b</i>) ⊗ st(<i>c</i>), where the sum is over the <i>m</i> + 1 ways to write <i>a</i> (considered as a sequence of <i>m</i> integers) as a concatenation of two sequences <i>b</i> and <i>c</i>, and st(<i>b</i>) is the standardization of <i>b</i>, where the elements of the sequence <i>b</i> are reduced to be a set of the form {1, 2, ..., <i>n</i>} while preserving their order. </p><p>The antipode has infinite order. </p> <h2 id="relation-to-other-algebras">Relation to other algebras</h2> <p>The Hopf algebra of permutations relates the rings of <a href="/facts/Symmetric_function/IB8N4T3m">symmetric functions</a>, <a href="/facts/Quasisymmetric_function/VSh6oTCV">quasisymmetric functions</a>, and <a href="/facts/Noncommutative_symmetric_function/lFFIJgNc">noncommutative symmetric functions</a>, (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. </p> <ul><li>Hazewinkel, Michiel; Gubareni, Nadiya; Kirichenko, V. V. (2010), <i>Algebras, rings and modules. Lie algebras and Hopf algebras</i>, Mathematical Surveys and Monographs, vol. 168, Providence, RI: American Mathematical Society, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-8218-5262-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2724822">2724822</a>, <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1211.16023">1211.16023</a></li> <li><a href="/facts/Claudia_Malvenuto/VBGxrxNz">Malvenuto, Claudia</a>; Reutenauer, Christophe (1995), "Duality between quasi-symmetric functions and the Solomon descent algebra", <i>J. Algebra</i>, 177 (3): 967–982, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1006%2Fjabr.1995.1336">10.1006/jabr.1995.1336</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1358493">1358493</a></li> <li>Poirier, Stéphane; Reutenauer, Christophe (1995), "Algèbres de Hopf de tableaux", <i>Ann. Sci. Math. Québec</i>, 19 (1): 79–90, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1334836">1334836</a></li></ul>