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.

124 questions
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Invariants in the ring of coinvariants

This seems to be part of folklore, but I can't find a proof anyqhere, only references to "a very nice proof of a special case by Arkady Berenstein in a seminar I attended in 1989". Okay, here it goes....
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Murnaghan-Nakayama rule for general dimension of a hook

Let $m,r\in \mathbb{N}, n=rm$. The character of symmetric group $\chi^{\lambda}$ where $\lambda$ is of the from $(k,1^{n-k})$ evaluated at the conjugacy class of $S_{n}$ of the from $(r,\ldots,r)$, ...
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How does the sum of closed walks change when a new vertex is added to a subgraph?

Suppose I calculate the sum of all closed walks from vertex i in a subgraph of some graph (the subgraph is composed of a subset of vertices from the graph and all edges between those vertices from the ...
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Is there a explicit formula for the number of Semi-standard Young Tableaux over $\{1,\dots,n\}$ for a given partition $\lambda$ and a given type $\mu$

I was given an exercise to give all SSYT over $\{1,\dots,12\}$ of shape $\lambda=(4,4,3,1)$ and type $\mu=(4,2,2,2,2,0,\dots,0)$. Now I was wondering if there is an formula to say something about the ...
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Counting monic irreducible polynomials of a particular degree without use of finite fields?

I know that every irreducible polynomial over $\mathbb F_p[x]$ of degree equal to the degree $m$ of a finite field $\mathbb F_{p^{m}}$ has a root in the field. Using this result we can count the ...
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What is the expected value of the product of the number of heads you get and the number of tails you get when you flip n coins? [closed]

Ex. If you get $h$ heads and $n-h$ tails the product would be $n(n-h)$ I want to know the expected value of this product.
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What do we mean when we say the Schur functions form a basis.

This has always bugged me. When we are examining symmetric functions (or polynomials if you prefer finitely many variables), we have an easy choice of basis with the monomial symmetric functions. As ...
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Reference request: Representation theory over fields of characteristic zero

Many representation theory textbooks and online resources work with the field of complex numbers or more generally algebraically closed fields of characteristic zero. I am looking for a good textbook ...
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Exercise on number of walks in a graph

The following is an exercise (Exercise #2 (a), Chapter 3, page 28) from Richard Stanley's Algebraic Combinatorics. Let $G$ be a finite graph (allowing loops and multiple edges). Suppose that ...
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Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...
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Proof that Paley Graphs are strongly regular with parameters $(p,\frac{p-1}{2},\frac{p-5}{4},\frac{p-1}{4})$

A Paley graph is strongly regular with parameters $(p,\frac{p-1}{2},\frac{p-5}{4},\frac{p-1}{4})$. I need to prove that, and obtain the parameters too. Proving it is regular valency $\frac{p-1}{2}$ is ...
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No triangles or rectangles in a Moore graph of diameter 2.

Can somebody explain why there cannot be any triangles or squares in a Moore graph with diameter 2? This was stated without proof in my class.
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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 ...
Orbital dimension of the action of $S_n$ on 2-subsets
I have a question on a proof in a paper on the orbital dimension of a permutation group. Let $G \le S^\Omega$ be a permutation group. A base for $G$ is a subset $\Sigma \subseteq \Omega$ for which ...