User:Richard Pinch/sandbox-3

Burnside ring

of a finite group $G$

For a $G$-set $X$, that is, a finite set with a group action by $G$, let $[X]$ denote the isomorphism class of $X$. These classes form a semi-ring with addition given by disjoint union $[X] + [Y] = [X \sqcup Y]$ and multiplication given by Cartesian product $[X] \cdot [Y] = [X \times Y]$. The zero element of this semiring is the (class of the) empty set and the multiplicative identity is the (class of the) one-point set. The Burnside ring $B(G)$ is the Grothendieck ring of this semiring.

The Burnside algebra of $G$ over a field $K$ is the $K$-algebra $K \otimes B(G)$. It is semi-simple if the characteristic of $K$ is zero or prime to the order of $G$.

References

• David D. Benson, Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras Cambridge University Press (1998) ISBN 0-521-63653-1 Zbl 0908.20001

Necklace algebra

The algebra $N(R)$ over a ring $R$ with additive group $R^{\mathbf{N}} = \{ a = (a_1,a_2,\ldots) : a_i \in R \}$ and multiplication given by $$ (a * b)_n = \sum_{i,j : \mathrm{lcm}(i,j) = m} \mathrm{hcf}(i,j) a_i b_j \ . $$ The definition of multiplication generalises the Metropolis–Rota multiplication formula for the necklace polynomials.

For a binomial ring $R$, the necklace algebra $N(R)$ is isomorphic to the universal lambda-ring $\Lambda(R)$ via $$ \prod_{n=1}^\infty \left({ 1 - (-t)^n }\right)^{a_n} \longleftrightarrow ( a_n ) \ , $$ which may be regarded as an abstraction of the Artin–Hasse exponential map. Here the exponentiation by elements $a \in R$ is defined by the binomial series $$ (1 + tx)^a = 1 + \sum_{n=1}^\infty \binom{a}{n} x^n t^n \ . $$

References

• Yau, Donald Lambda-rings World Scientific (2010) ISBN 978-981-4299-09-1 Zbl 1198.13003

Necklace polynomial

A polynomial of the form $$ M_n(x) = \frac{1}{n} \sum_{d | n} \mu(d) x^{n/d} \ . $$ Here $\mu$ is the Möbius function. When $x$ is a natural number, $M_n(x)$ counts the number of "necklaces": assignments of $n$ colours to $x$ beads under cyclic symmetry which are "primitive", that is, not the repetition of a proper subsequence of colour assignments.

Metropolis and Rota showed that $$ M_n(xy) = \sum_{[i,j]=n} (i,j) M_i(x) M_j(y) $$ where $[,]$ denotes least common multiple and $(,)$ highest common factor.

References

• Yau, Donald Lambda-rings World Scientific (2010) ISBN 978-981-4299-09-1 Zbl 1198.13003