The term ‘shuffle’ conjures up the idea of shuffling a pack of cards. In fact the mathematical idea is nearer to shuffling two packs of cards one through the other. Suppose we have a pack of cards and a pack of cards and we build a pack of cards, whilst retaining the order on the two ‘sub-packs’. The result is a -shuffle.
(\mu_1, \cdots, \mu_p, \nu_1, \cdots, \nu_q)
of subject to the condition that
\mu_1 \lt \mu_2 \lt \cdots \lt \mu_p
\nu_1 \lt \nu_2 \lt \cdots \lt \nu_q \,.
The signature of a -shuffle is the signature of the corresponding permutation.
The same concept viewed from the other end leads to unshuffles . These are just shuffles but are used in dual situations in the applications. The definition that follows is ‘from the literature’. It is equivalent to that of shuffle that we gave above. (Although not needed, it is important to note the different terminology used in certain applications of the idea for when original source material is consulted.)
We say that a permutation is a -unshuffle, if and .
You can also say that is a -unshuffle if when .
Related to the product of simplices: shuffles control the Eilenberg-Zilber map. See there for details.
Shuffles are used in defining the pre-cgc structure on in the theory of differential graded coalgebras
Shuffles are also used for defining the shuffle product on , see differential graded Hopf algebra.