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=Burnside ring=
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''of a finite group $G$''
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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.
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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 $G$.
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====References====
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* 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}}
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=Necklace algebra=
 
=Necklace algebra=
The algebra over a ring $R$ with additive group $R^{\mathbf{N}} = \{ a = (a_1,a_2,\ldots) : a_i \in R \}$ and multiplication given by
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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 \ .
 
(a * b)_n = \sum_{i,j : \mathrm{lcm}(i,j) = m} \mathrm{hcf}(i,j) a_i b_j \ .
 
$$
 
$$
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The definition of multiplication generalises the Metropolis–Rota multiplication formula for the [[necklace polynomial]]s.
  
=Harmonic number=
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For a [[binomial ring]] $R$, the necklace algebra $N(R)$ is isomorphic to the universal [[lambda-ring]] $\Lambda(R)$ via
Commonly, a partial sum of the [[harmonic series]]
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$$
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\prod_{n=1}^\infty \left({ 1 - (-t)^n }\right)^{a_n} \longleftrightarrow ( a_n )  \ ,
 
$$
 
$$
H_n = \sum_{k=1}^n \frac{1}{k} \ .
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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]]
 
$$
 
$$
A generalised harmonic number is a partial num of the zeta function
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(1 + tx)^a = 1 + \sum_{n=1}^\infty \binom{a}{n} x^n t^n  \ .
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$$
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====References====
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* Yau, Donald ''Lambda-rings'' World Scientific (2010) ISBN 978-981-4299-09-1 {{ZBL|1198.13003}}
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=Necklace polynomial=
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A polynomial of the form 
 
$$
 
$$
H_n^{(s)} = \sum_{k=1}^n \frac{1}{k^s} \ .
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M_n(x) = \frac{1}{n} \sum_{d | n} \mu(d) x^{n/d} \ .
 
$$
 
$$
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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.
  
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 divisors|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
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Metropolis and Rota showed that
 
$$
 
$$
1,\ 6,\ 28,\ 140,\ 270,\ 496,\ 672 \ .
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M_n(xy) = \sum_{[i,j]=n} (i,j) M_i(x) M_j(y)
 
$$
 
$$
An even [[perfect number]] is a harmonic number.
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where $[,]$ denotes [[least common multiple]] and $(,)$ [[highest common factor]].
  
 
====References====
 
====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
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* Yau, Donald ''Lambda-rings'' World Scientific (2010) ISBN 978-981-4299-09-1 {{ZBL|1198.13003}}
* 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
 

Latest revision as of 14:03, 21 January 2021

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
How to Cite This Entry:
Richard Pinch/sandbox-3. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-3&oldid=37681