{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T20:10:54Z","timestamp":1772136654463,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.37236\/14424","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:43Z","timestamp":1772135023000},"source":"Crossref","is-referenced-by-count":0,"title":["The Lie Algebra $\\mathfrak{sl}_4(\\mathbb C)$  and the Hypercubes"],"prefix":"10.37236","volume":"33","author":[{"given":"William J.","family":"Martin","sequence":"first","affiliation":[]},{"given":"Paul","family":"Terwilliger","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2026,2,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v33i1p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T19:43:43Z","timestamp":1772135023000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v33i1p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1,9]]}},"URL":"https:\/\/doi.org\/10.37236\/14424","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,13]]},"article-number":"P1.29"}}