Necklace ring

In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.

Definition

If A is a commutative ring then the necklace ring over A consists of all infinite sequences [math]\displaystyle{ (a_1, a_2, ...) }[/math] of elements of A. Addition in the necklace ring is given by pointwise addition of sequences. Multiplication is given by a sort of arithmetic convolution: the product of [math]\displaystyle{ (a_1, a_2, ...) }[/math] and [math]\displaystyle{ (b_1, b_2, ...) }[/math] has components

[math]\displaystyle{ \displaystyle c_n=\sum_{[i,j]=n}(i,j)a_ib_j }[/math]

where [math]\displaystyle{ [i,j] }[/math] is the least common multiple of [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math], and [math]\displaystyle{ (i,j) }[/math] is their greatest common divisor.

This ring structure is isomorphic to the multiplication of formal power series written in "necklace coordinates": that is, identifying an integer sequence [math]\displaystyle{ (a_1, a_2, ...) }[/math] with the power series [math]\displaystyle{ \textstyle\prod_{n\geq 0} (1{-}t^n)^{-a_n} }[/math].

See also

• Witt vector

References

• Hazewinkel, Michiel (2009). "Witt vectors I". Handbook of Algebra. 6. Elsevier/North-Holland. pp. 319–472. ISBN 978-0-444-53257-2. Bibcode: 2008arXiv0804.3888H. 
• Metropolis, N.; Rota, Gian-Carlo (1983). "Witt vectors and the algebra of necklaces". Advances in Mathematics 50 (2): 95–125. doi:10.1016/0001-8708(83)90035-X.