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Necklace ring

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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 ( a 1 , a 2 , . . . ) {\displaystyle (a_{1},a_{2},...)} 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 ( a 1 , a 2 , . . . ) {\displaystyle (a_{1},a_{2},...)} and ( b 1 , b 2 , . . . ) {\displaystyle (b_{1},b_{2},...)} has components

c n = ∑ [ i , j ] = n ( i , j ) a i b j {\displaystyle \displaystyle c_{n}=\sum _{=n}(i,j)a_{i}b_{j}}

where [ i , j ] {\displaystyle } is the least common multiple of i {\displaystyle i} and j {\displaystyle j} , and ( i , j ) {\displaystyle (i,j)} 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 ( a 1 , a 2 , . . . ) {\displaystyle (a_{1},a_{2},...)} with the power series ∏ n ≥ 0 ( 1 − t n ) − a n {\displaystyle \textstyle \prod _{n\geq 0}(1{-}t^{n})^{-a_{n}}} .

See also

References

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