All Questions
Tagged with algebraic-combinatorics abstract-algebra
9
questions
6
votes
1
answer
102
views
A Question on the Pedagogical Logic Behind the Order of Two Given Exercises
In Lang's Algebra, the following two exercises are presented to the reader in the following order:
Groups Exercise 15: Let $G$ be a finite group acting on $S$, a finite set of at least $2$ elements. ...
- 151
3
votes
1
answer
82
views
Given a transitive and faithful permutation group $G$, is each set of syntactically transitive permutations connected by another permutation in $G$?
$G$ is a permutation group of degree $n \geq 4$ which acts transitively and faithfully on a set $X$ with $|X| = n$.
Given indices $i < j < k \leq n$, elements $\alpha \neq \beta \neq \gamma \in ...
- 317
3
votes
0
answers
99
views
Representation for the Bose-Mesner algebra and its dual.
If you are familiar with the algebra, just skip the following brief introduction, that is fine.
In the study of Bose-Mesner algebra. We know given that the commutative association scheme $\mathfrak{X}...
- 1,366
4
votes
0
answers
118
views
Constant sum of characters
Let $q$ be a prime power and $\omega=\exp(2\pi i/q)$. For a fixed $y\in\mathbb{Z}_q^n$, the map
$$\mathbb{Z}_q^n\ni x\mapsto \omega^{x\cdot y}=\omega^{x_1y_1+\dots+x_ny_n}$$
is a character of $\...
- 450
3
votes
1
answer
184
views
To show two formal power series equal
I am wondering whether the following two formal power series are equal:
$A(x)=\Pi_{k=1}^{\infty}\frac{1}{1-x^{2k-1}}$, $B(x)=\Pi_{k=1}^{\infty}(1+x^k)$.
- 454
1
vote
0
answers
316
views
Necklace polynomial recurrence relation
Let the number of monic, irreducible polynomials of degree $n$ over $F_q$ be $f(n)$, then $f(1)=q$, and to calculate $f(n)$, I would count the number of monic, reducible polynomials of degree $n$ this ...
- 3,806
1
vote
1
answer
169
views
prove a polynomial identity..
The equation is that
$h_m(x_1, \cdots, x_n, a)-h_m(x_1, \cdots, x_n, b)=(a-b)h_{m-1}(x_1, \cdots, x_n, a, b)$ where $h_m$ is a complete homogeneous symmetric polynomial.
See and find several ...
- 313
3
votes
2
answers
738
views
Expressions for symmetric power sums in terms of lower symmetric power sums
The Newton symmetric power sums $p_k(x_1, \ldots, x_n)$, for $k \geq 1$, are given by
$$
p_k(x_1, \ldots, x_n) = \sum_{i=1}^n x_i^k.
$$
Do you know if it's possible to express $p_k$ in terms of (non-...
- 2,003
2
votes
0
answers
130
views
Proving Crapo's Lemma
Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to ...
- 9,205