In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product X ⧢ Y of two words X, Y: the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation.
The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative.
The shuffle product of words of lengths m and n is a sum over the (m+n)!/m!n! ways of interleaving the two words, as shown in the following examples:
- ab ⧢ xy = abxy + axby + xaby + axyb + xayb + xyab
- aaa ⧢ aa = 10aaaaa
It may be defined inductively by
- u ⧢ ε = ε ⧢ u = u
- ua ⧢ vb = (u ⧢ vb)a + (ua ⧢ v)b
where ε is the empty word, a and b are single elements, and u and v are arbitrary words.
The shuffle product was introduced by Eilenberg & Mac Lane (1953). The name "shuffle product" refers to the fact that the product can be thought of as a sum over all ways of riffle shuffling two words together: this is the riffle shuffle permutation. The product is commutative and associative.
The closely related infiltration product was introduced by Chen, Fox & Lyndon (1958). It is defined inductively on words over an alphabet A by
- fa ↑ ga = (f ↑ ga)a + (fa ↑ g)a + (f ↑ g)a
- fa ↑ gb = (f ↑ gb)a + (fa ↑ g)b
- ab ↑ ab = ab + 2aab + 2abb + 4 aabb + 2abab
- ab ↑ ba = aba + bab + abab + 2abba + 2baab + baba
The infiltration product is also commutative and associative.
- Lothaire (1997) pp.101,126
- Lothaire (1997) p.126
- Lothaire (1997) p.128
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- Eilenberg, Samuel; Mac Lane, Saunders (1953), "On the groups of H(Π,n). I", Annals of Mathematics, Second Series, 58 (1): 55–106, doi:10.2307/1969820, ISSN 0003-486X, JSTOR 1969820, MR 0056295, Zbl 0050.39304
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