The Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercubes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Martin, William J. Terwilliger, Paul |
| author_facet | Martin, William J. Terwilliger, Paul |
| contents | We describe a relationship between the Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercube graphs. Consider the $\mathbb C$-algebra $P$ of polynomials in four commuting variables. We turn $P$ into an $\mathfrak{sl}_4(\mathbb C)$-module on which each element of $\mathfrak{sl}_4(\mathbb C)$ acts as a derivation. Then $P$ becomes a direct sum of irreducible $\mathfrak{sl}_4(\mathbb C)$-modules
$P = \sum_{N\in \mathbb N} P_N$, where $P_N$ is the $N$th homogeneous component of $P$. For $N\in \mathbb N$ we construct some additional $\mathfrak{sl}_4(\mathbb C)$-modules ${\rm Fix}(G)$ and $T$. For these modules the underlying vector space is described as follows. Let $X$ denote the vertex set of the hypercube $H(N,2)$, and let $V$ denote the $\mathbb C$-vector space with basis $X$. For the automorphism group $G$ of $H(N,2)$, the action of $G$ on $X$ turns $V$ into a $G$-module. The vector space $V^{\otimes 3} = V \otimes V \otimes V$ becomes a $G$-module such that $g(u \otimes v \otimes w)= g(u) \otimes g(v) \otimes g(w)$ for $g\in G$ and $u,v,w \in V$. The subspace ${\rm Fix}(G)$
of $V^{\otimes 3}$ consists of the vectors in $V^{\otimes 3}$ that are fixed by every element in $G$.
Pick $\varkappa \in X$. The corresponding subconstituent algebra $T$ of $H(N,2)$ is the subalgebra of ${\rm End}(V)$ generated by the adjacency map $\sf A$ of $H(N,2)$ and the dual adjacency map ${\sf A}^*$ of $H(N,2)$ with respect to $\varkappa$. In our main results, we turn ${\rm Fix}(G)$ and $T$ into $\mathfrak{sl}_4(\mathbb C)$-modules, and display $\mathfrak{sl}_4(\mathbb C)$-module isomorphisms $P_N \to {\rm Fix}(G) \to T$.
We describe the $\mathfrak{sl}_4(\mathbb C)$-modules $P_N$, ${\rm Fix}(G)$, $T$ from multiple points of view. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03951 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercubes Martin, William J. Terwilliger, Paul Combinatorics Representation Theory 05E30 We describe a relationship between the Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercube graphs. Consider the $\mathbb C$-algebra $P$ of polynomials in four commuting variables. We turn $P$ into an $\mathfrak{sl}_4(\mathbb C)$-module on which each element of $\mathfrak{sl}_4(\mathbb C)$ acts as a derivation. Then $P$ becomes a direct sum of irreducible $\mathfrak{sl}_4(\mathbb C)$-modules $P = \sum_{N\in \mathbb N} P_N$, where $P_N$ is the $N$th homogeneous component of $P$. For $N\in \mathbb N$ we construct some additional $\mathfrak{sl}_4(\mathbb C)$-modules ${\rm Fix}(G)$ and $T$. For these modules the underlying vector space is described as follows. Let $X$ denote the vertex set of the hypercube $H(N,2)$, and let $V$ denote the $\mathbb C$-vector space with basis $X$. For the automorphism group $G$ of $H(N,2)$, the action of $G$ on $X$ turns $V$ into a $G$-module. The vector space $V^{\otimes 3} = V \otimes V \otimes V$ becomes a $G$-module such that $g(u \otimes v \otimes w)= g(u) \otimes g(v) \otimes g(w)$ for $g\in G$ and $u,v,w \in V$. The subspace ${\rm Fix}(G)$ of $V^{\otimes 3}$ consists of the vectors in $V^{\otimes 3}$ that are fixed by every element in $G$. Pick $\varkappa \in X$. The corresponding subconstituent algebra $T$ of $H(N,2)$ is the subalgebra of ${\rm End}(V)$ generated by the adjacency map $\sf A$ of $H(N,2)$ and the dual adjacency map ${\sf A}^*$ of $H(N,2)$ with respect to $\varkappa$. In our main results, we turn ${\rm Fix}(G)$ and $T$ into $\mathfrak{sl}_4(\mathbb C)$-modules, and display $\mathfrak{sl}_4(\mathbb C)$-module isomorphisms $P_N \to {\rm Fix}(G) \to T$. We describe the $\mathfrak{sl}_4(\mathbb C)$-modules $P_N$, ${\rm Fix}(G)$, $T$ from multiple points of view. |
| title | The Lie algebra $\mathfrak{sl}_4(\mathbb C)$ and the hypercubes |
| topic | Combinatorics Representation Theory 05E30 |
| url | https://arxiv.org/abs/2505.03951 |