User:Richard Pinch/sandbox-3

Burnside ring

of a group $G$

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

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

Binomial ring

A ring $R$ with torion-free addition in which all binomial symbols $$ \binom{x}{n} = \frac{x(x-1)\cdots(x-n+1)}{n!} $$ are well-defined as polynomials in $R[x]$. Clearly any field of characteristic zero is a binomial ring, as is the ring of integers $\mathbf{Z}$.

If $R$ is binomial, then defining $\lambda^n (a) = \binom{a}{n}$ makes $R$ a lambda-ring with operators $\lambda^n$. The Adams operations are all equal to the identity.

References

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

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.

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

Harmonic number

Commonly, a partial sum of the harmonic series $$ H_n = \sum_{k=1}^n \frac{1}{k} \ . $$ A generalised harmonic number is a partial num of the zeta function $$ H_n^{(s)} = \sum_{k=1}^n \frac{1}{k^s} \ . $$

However, Pomerance has defined a harmonic number to be a natural number $n$ for which the harmonic mean of the divisors of $n$ is an integer; equivalently $\sigma(n)$ divides $n.d(n)$ where $\sigma(n)$ is the sum of the divisors of $n$ and $d(n)$ is the number of divisors: these are also called Øre numbers. The first seven such numbers are $$ 1,\ 6,\ 28,\ 140,\ 270,\ 496,\ 672 \ . $$ An even perfect number is a harmonic number.

References

• Guy, Richard K. "Almost Perfect, Quasi-Perfect, Pseudoperfect, Harmonic, Weird, Multiperfect and Hyperperfect Numbers". Unsolved Problems in Number Theory (2nd ed.). New York: Springer-Verlag (1994). pp. 16, 45–53
• Milovanović, Gradimir V., Rassias, Michael Th. (edd.) Analytic Number Theory, Approximation Theory, and Special Functions: In Honor of Hari M. Srivastava Springer (2014) ISBN 149390258X
• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, edd. Handbook of number theory I. Dordrecht: Springer-Verlag (2006). ISBN 1-4020-4215-9. Zbl 1151.11300
• Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). ISBN 1-4020-2546-7. Zbl 1079.11001
• Wagstaff, Samuel S. The Joy of Factoring Student mathematical library 68 American Mathematical Society (2013) ISBN 1470410486