All Questions
Tagged with algebraic-combinatorics determinant
8
questions
0
votes
0
answers
38
views
Are there Plucker-like relations for the tensor product of two decomposable differential forms?
Let $V$ be a $k$-dimensional vector space, and consider a decomposable tensor in $\bigwedge^\ell V \otimes \bigwedge^m V$ having the form
$$
\mathbf{v} \otimes \mathbf{w} := v_1 \wedge \cdots \wedge ...
- 127
1
vote
0
answers
39
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Evaluating character functions
Immanants generalize the notion of determinant and permanent, and it is defined as $$Imm_{\lambda}(A)=\sum_{\sigma \in S_n}\chi_{\lambda}(\sigma)\prod_{i=1}^{n}a_{i,\sigma(i)}$$
where $\chi_{\lambda}$ ...
- 125
2
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34
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Determinant of $(p(x_i-x_j))_{1 \leq i,j \leq n}$ for polynomial $p(x)$.
Let $p(x) = a_0 + a_1 x + \ldots + a_k x^k$ be a polynomial.
Can anything be said in general about the determinant $\mathrm{det}_{1 \leq i,j \leq n} (p(x_i-x_j))$ for a collection of variables $x_1,\...
- 674
5
votes
2
answers
573
views
Paths must cross in Lindström-Gessel-Viennot on the lattice
The following question (which I am going to answer myself) serves
to close a little gap in some combinatorial proofs that use the
Lindström--Gessel--Viennot lemma. Namely, I will show a little lemma,
...
- 17.9k
2
votes
0
answers
82
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About the determinant of a symmetric matrix with even diagonal
Let $A$,$B$ two integer, symmetric, $n\times n$ matrix with even diagonal. We suppose also that $A$ has odd determinant (so $n$ is even).
Even discarding the hypothesis of even diagonal for $A$ it'...
- 674
4
votes
3
answers
742
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Graph Theory - Application of Kirchoff's Matrix Tree Theorem
Calculate the number of spanning trees of the graph that you obtain by removing one edge from $K_n$.
(Hint: How many of the spanning trees of $K_n$ contain the edge?)
I know the number is $(n-2)n^{...
- 41
3
votes
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44
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Calculating determinants
Let $n\geq 2$ be an integer and let $\Sigma$ be the collection of all $2$-subsets (a 2-set is a set that contains $2$ elements) of $[n]=\{1,2,\dots,n\}$, thus $\Sigma$ contains $\binom{n}{2}$ elements....
- 4,865
1
vote
0
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876
views
Pfaffian And Determinant
I am working in tilings using Pfaffian. There is a basic property, namely:
Let $ B$ be a $n\times n$ matrix. Let
$$ A = \begin{pmatrix}
0 & B\\
-B^T & 0
\end{pmatrix}$$
then
$$\...
- 253