Answer by Timothy Chow for Applications of Representation Theory in...
Here's an example that involves a group other than the symmetric group.Let ${\mathbb F}_{q^n}$ denote the finite field with $q^n$ elements. Since ${\mathbb F}_{q^n}$ is a vector space of dimension $n$...
View ArticleAnswer by Mark Wildon for Applications of Representation Theory in Combinatorics
Define a partition move to be the removal and then addition of a box of the Young diagram of a partition. Thus there are unique partition moves from $(2,1)$ to $(3)$ and from $(2,1)$ to $(1,1,1)$, and...
View ArticleAnswer by John Murray for Applications of Representation Theory in Combinatorics
It seems to me that all answers to this question will involve the symmetric group. Here is one example from the representation theory of $S_n$ over a field of characteristic...
View ArticleAnswer by Tobias Kildetoft for Applications of Representation Theory in...
The Robinson-Schensted correspondence has already been mentioned in another answer, as providing a bijection between permutations of $\{1,\dots,n\}$ and pairs of standard Young tableaux of the same...
View ArticleAnswer by Allen Knutson for Applications of Representation Theory in...
Maschke tells you that $\mathbb C[G] \cong \bigoplus_V V^*\otimes V$, where $V$ runs over the irreps of a finite group $G$. Apply this to $G= S_n$ and take dimensions, and you get $n! = \sum_{\lambda...
View ArticleAnswer by Gro-Tsen for Applications of Representation Theory in Combinatorics
You may be interested in Yuval Filmus's survey paper Spectral Methods for Intersection Problems (Friedgut's Research Program in Extremal Combinatorics) (and/or his thesis which seems to have exactly...
View ArticleAnswer by Fedor Petrov for Applications of Representation Theory in...
Given $k$ conjugacy classes $C_1,C_2,\dots,C_k$ in a finite group $G$ with unity $e$, we may ask what is the probability $p(C_1,\dots,C_k)$ that $g_1\dots g_k=e$ if $g_i\in C_i$ is chosen uniformly at...
View ArticleAnswer by Zach H for Applications of Representation Theory in Combinatorics
Many plane partition enumeration formulae have proofs using representation theory. For example, let $PP((p)^q,m)$ denote the $p \times q$ plane partitions of height at most $m$. For $p > q$, let...
View ArticleAnswer by Igor Pak for Applications of Representation Theory in Combinatorics
Let$$(\ast) \qquad g(\alpha,\beta,\gamma) := \sum_{\sigma,\omega,\pi} \text{sign}(\sigma\omega\pi) \, C(\alpha-\sigma,\beta-\omega,\gamma-\pi),$$where $\alpha,\beta,\gamma \vdash n$ are partitions with...
View ArticleAnswer by Asaf for Applications of Representation Theory in Combinatorics
One of the most well known applications of representation theory in combinatorics is the explicit construction of Expander graphs by Margulis (using Kazhdan's property (T)). See any book about...
View ArticleAnswer by Ofir Gorodetsky for Applications of Representation Theory in...
Alon and Kozma re-proved Kirchoff’s matrix-tree theorem using representation theory of $S_n$:Let $G$ be any weighted graph, and denote by $w_e$ the weight of the edge $e$. For a spanning tree $T$,...
View ArticleAnswer by Per Alexandersson for Applications of Representation Theory in...
Many identities involving multiplicative structure constants originate in representation theory. There are plenty of positivity statements about such structure constants but no combinatorial proof is...
View ArticleAnswer by Richard Stanley for Applications of Representation Theory in...
It is unclear exactly what is meant by a "combinatorial identity." Forinstance, let $X_n$ denote the set of alternating permutations in thesymmetric group $\mathfrak{S}_n$ that are also cycles of...
View ArticleAnswer by NietzscheanAI for Applications of Representation Theory in...
The FFT on permutations is intimately connected with the representation of the Symmetric group:See"Group theoretical methods in Machine Learning" by Risi Kondor.
View ArticleApplications of Representation Theory in Combinatorics
What are the examples of interesting combinatorial identities (e.g. bijection between two sets of combinatorial objects) that can be proved using representation theory, or has some representation...
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