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Ring of symmetric functions
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<p>In <a href="/facts/Algebra/nMdqczjo">algebra</a> and in particular in <a href="/facts/Algebraic_combinatorics/9b47n3Fh">algebraic combinatorics</a>, the ring of symmetric functions is a specific limit of the <a href="/facts/Ring_(mathematics)/JnCUXz63">rings</a> of <a href="/facts/Symmetric_polynomial/GNhguVcn">symmetric polynomials</a> in <i>n</i> indeterminates, as <i>n</i> goes to infinity. This ring serves as universal structure in which relations between symmetric polynomials can be expressed in a way independent of the number <i>n</i> of indeterminates (but its elements are neither polynomials nor functions). Among other things, this ring plays an important role in the <a href="/facts/Representation_theory_of_the_symmetric_group/ht0T7JNt">representation theory of the symmetric group</a>. </p><p>The ring of symmetric functions can be given a <a href="/facts/Coproduct/0bJ8SCdu">coproduct</a> and a <a href="/facts/Bilinear_form/gyvzn9ym">bilinear form</a> making it into a positive selfadjoint <a href="/facts/Graded_algebra/AIUjkSaP">graded</a> <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a> that is both commutative and cocommutative. </p>