• Source: 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



(

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 _{[i,j]=n}(i,j)a_{i}b_{j}}


where



[
i
,
j
]


{\displaystyle [i,j]}

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


Witt vector




References


Hazewinkel, Michiel (2009). "Witt vectors I". Handbook of Algebra. Vol. 6. Elsevier/North-Holland. pp. 319–472. arXiv:0804.3888. Bibcode:2008arXiv0804.3888H. ISBN 978-0-444-53257-2. MR 2553661.
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. MR 0723197.

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