Questions tagged [algebraic-combinatorics]
For problems involving algebraic methods in combinatorics (especially group theory and representation theory) as well as combinatorial methods in abstract algebra.
265
questions
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Is there anywhere some explicit Bruhat decompositions are written down?
Question in title: most places I see Bruhat decompositions treated they're only briefly mentioned and no examples are given.
Also, I calculated the following regarding the Bruhat decomposition of $\...
- 24.9k
14
votes
6
answers
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views
Combinatorial interpretation of an alternating binomial sum
Let $n$ be a fixed natural number. I have reason to believe that $$\sum_{i=k}^n (-1)^{i-k} \binom{i}{k} \binom{n+1}{i+1}=1$$ for all $0\leq k \leq n.$ However I can not prove this. Any method to prove ...
- 547
1
vote
0
answers
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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
$$\...
- 253
3
votes
1
answer
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Prove that $h_r(x_1,\dots,x_n)=\sum^n_{k=1}x^{n-1+r}_k\prod_{i\neq k}(x_k-x_i)^{-1}$
Let $h_r\left(x_1,\ldots,x_n\right)$ denote the $r$-th complete homogeneous symmetric polynomial in the $n$ indeterminates $x_1, \ldots, x_n$ (that is, the sum of all degree-$r$ monomials in these $n$ ...
- 31
-1
votes
2
answers
609
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Young tableaux of shape lambda. [closed]
Consider the partition $\lambda=(m,n-m)$ of $n$ (thus $2m \ge n$).
The number of standard Young
tableaux of shape $\lambda$ is given by
$$f_{(m,n-m)} = \binom nm - \binom{n}{m+1}$$
a) Prove this ...
- 1,302
20
votes
3
answers
5k
views
Hartshorne Problem 1.2.14 on Segre Embedding
This is a problem in Hartshorne concerning showing that the image of $\Bbb{P}^n \times \Bbb{P}^m$ under the Segre embedding $\psi$ is actually irreducible. Now I have shown with some effort that $\psi(...
user38268
3
votes
0
answers
234
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Symmetrization of Powersum polynomials
Let $n\in\mathbb{N}$. Then, for $i\in\mathbb{N}$, the $i-$th power sum if defined to be the polynomial $p_i^{(n)}:=\sum_{j=1}^n x_j^i$ in $n$ indeterminates $x_1,x_2,\ldots,x_n$.
Then let $\lambda:=(\...
- 51
8
votes
2
answers
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Given a list of $2^n$ nonzero vectors in $GF(2^n)$, do some $2^{n-1}$ of them sum to 0?
Let $G=(\mathbb{Z/2Z})^n$ written additively, $n>1$. (you can think of it as $\mathbb{F}_{2^n}$ but I didn't find that useful... yet)
Let $v_i$ be nonzero elements of $G$ for $i \in \{1 \dots 2^n \...
- 786
15
votes
2
answers
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Identity involving partitions of even and odd parts.
First, denote by $p_E(n)$ be the number of partitions of $n$ with an even number of parts, and let $p_O(n)$ be those with an odd number of parts. Moreover, let $p_{DO}(x)$ be the number of partitions ...
- 153
7
votes
0
answers
350
views
Uses of Chevalley-Warning
In the recent IMC 2011, the last problem of the 1st day (no. 5, the hardest of that day) was as follows:
We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The problem asks :
Prove the ...
- 6,265