The pancake graph Pn is the Cayley graph of the symmetric group Sn on n elements generated by prefix reversals. Pn has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is (n − 1)-regular, vertex-transitive, and one can embed cycles in it of length ℓ with 6 ≤ ℓ ≤ n!. The burnt pancake graph BPn, which is the Cayley graph of the group of signed permutations Bn using prefix reversals as generators, has similar properties. Indeed, BPn is n-regular and vertextransitive. In this paper, we show that BPn has every cycle of length ℓ with 8 ≤ ℓ ≤ 2 n n!. The proof given is a constructive one that utilizes the recursive structure of BPn.We also present a complete characterization of all the 8-cycles in BPn for n ≥ 2, which are the smallest cycles embeddable in BPn, by presenting their canonical forms as products of the prefix reversal generators.
Communicated by R. Vakil MSC: 20C30 20C08 15A15 17B37 a b s t r a c t We use Kazhdan-Lusztig polynomials and subspaces of the polynomial ring C[x 1,1 , . . . , x n,n ] to give a new construction of the Kazhdan-Lusztig representations of S n . This construction produces exactly the same modules as those which Clausen constructed using a different basis in [M. Clausen, Multivariate polynomials, standard tableaux, and representations of symmetric groups, J. Symbolic Comput. (11), 5-6 (1991) 483-522. Invariant-theoretic algorithms in geometry (Minneapolis, MN, 1987)], and does not employ the Kazhdan-Lusztig preorders. We show that the two resulting matrix representations are related by a unitriangular transition matrix. This provides a C[x 1,1 , . . . , x n,n ]-analog of results due to Garsia and McLarnan, and McDonough and Pallikaros, who related the Kazhdan-Lusztig representations to Young's natural representations.
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We use a quantum analog of the polynomial ring $\mathbb{Z}[x_{1,1},\ldots, x_{n,n}]$ to modify the Kazhdan-Lusztig construction of irreducible $H_n(q)$-modules. This modified construction produces exactly the same matrices as the original construction in [$\textit{Invent. Math.}$ $\textbf{53}$ (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum polynomial ring.
Nous utilisons un analogue quantique de l'anneau $\mathbb{Z}[x_{1,1},\ldots,x_{n,n}]$ pour modifier la construction Kazhdan-Lusztig des modules-$H_n(q)$ irréductibles. Cette construction modifiée produit exactement les mêmes matrices que la construction originale dans [$\textit{Invent. Math.}$ $\textbf{53}$ (1979)], mais sans employer les préordres de Kazhdan-Lusztig. Notre résultat principal dépend de nouveaux résultats de disparition pour des immanants dans l'anneau polynôme de quantique.
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