Harmonics on posets.

*(English)*Zbl 0573.06001If P is a finite poset with maximal elements \(X_ N\), then for certain posets it is possible to decompose \(L^ 2(X_ N)=\oplus^{N}_{n=0}Harm(n)| X_ N\), where \(L^ 2(X_ N)\) is the set of complex valued functions defined on \(X_ N\) acted on by the automorphism group G of P via the permutation representation induced from the stabilizer H of a fixed element \(x^{(0)}\) of \(X_ N\), so that the components \(Harm(n)| X_ n\) are irreducible G-modules with unique right H-invariant functions \(f_ n\) such that \(f_ n(x^{(0)})=1\). Under the conditions given below, the decomposition of \(L^ 2(_ N)\) is multiplicity free, and the principal harmonics \(f_ n\) are orthogonal polynomials. The conditions derived by the author include well-known examples (classical maybe) among the great variety made possible by his construction. The theorem itself is very nice in its surprising cleanness for the class under discussion and should prove useful in many applications.

For 2N\(\leq v\), the poset P of all subspaces of dimension at most N of a v-dimensional vector space V over GF(q) ordered by set inclusion is a finite ranked poset with unique minimal element 0 which satisfies the following conditions: (A.1) P is a meet semilattice; (A.2) which is lower semi-modular; (A.3) whose automorphism group G acts transitively on \(X_ N\), the set of maximal elements and such that for \(x^{(0)}\in X_ N\), (A.4) the H-orbits on \(X_ n\), the elements of rank n, are \(\Omega_{ni}=\{\alpha \in X_ n|\) \(\alpha \wedge x^{(0)}\in X_{n-i}\}\), \(0\leq i\leq n\). Along with other examples, the class of posets satisfying these conditions includes a certain type of lexicographic sum \(P\cdot Q\) obtained by attaching copies of Q to the vertices of \(X_ N\), the maximal elements of P. If P satisfies (A.1) and (A.2), then \(d(x,y)=N-rank(x\wedge y)\) is a metric on \(X_ N\times X_ N.\)

Letting \(f^{\alpha}(\beta)=1\) if \(\alpha\leq \beta\), 0 otherwise, then \(\{f^{\alpha}:\) \(\alpha\in P\}\) is a basis for \(L^ 2(P)\). If Hom(n) is the collection of complex valued functions defined on \(P=span\{f^{\alpha}:\) \(\alpha \in X_ n\}\) and if \(Harm(n)=Hom(n)\wedge Ker d\), where \(df^{\alpha}=\sum_{\beta}f^{\beta}\), \(\alpha\) covers \(\beta\), then conditions (A.1)-(A.4) are sufficient to insure that the restrictions of the functions in Harm(n) to \(X_ N\) yield irreducible G-modules, and that the representation of G on \(L^ 2(X_ N)\) is multiplicity free with the decomposition \(L^ 2(X_ N)=\oplus^{N}_{n=0}Harm(n)|_{X_ N}\). Moreover, the normalized ”spherical” function for \(Harm(n)|_{X_ N}\) is \(f_ n(\Omega_{nj})=\sum^{n}_{i=0}c_ iF_{ni}(\Omega_{Nj})/d_ n\), where \(F_{ni}=\sum_{\alpha \in \Omega_{ni}}f^{\alpha}\) and \(\{F_{ni}:\) \(0\leq i\leq n\}\) is a basis for the right H-invariant functions in Hom(n). Also \(dF_{ni}=a_{n-1,i}F_{n-1,i}+b_{n- 1,i}F_{n-1,i-1}\), where \(a_{n-1,i}=| \{\gamma \in \Omega_{ni}:\) \(\gamma >\alpha \}|\), \(b_{n-1,i}=| \gamma \in \Omega_{n,i+1}:\) \(\gamma >\alpha \}| \neq 0\) for some \(a\in \Omega_{n-1,j}\), \((-1)^ ic_ i(\prod^{i-1}_{j=0}a_{n- 1,j}/b_{n-1,j})c_ 0\), and \(d_ n=| \{\alpha \in X_ n:\) \(\alpha \leq x^{(0)}\}|\).

For 2N\(\leq v\), the poset P of all subspaces of dimension at most N of a v-dimensional vector space V over GF(q) ordered by set inclusion is a finite ranked poset with unique minimal element 0 which satisfies the following conditions: (A.1) P is a meet semilattice; (A.2) which is lower semi-modular; (A.3) whose automorphism group G acts transitively on \(X_ N\), the set of maximal elements and such that for \(x^{(0)}\in X_ N\), (A.4) the H-orbits on \(X_ n\), the elements of rank n, are \(\Omega_{ni}=\{\alpha \in X_ n|\) \(\alpha \wedge x^{(0)}\in X_{n-i}\}\), \(0\leq i\leq n\). Along with other examples, the class of posets satisfying these conditions includes a certain type of lexicographic sum \(P\cdot Q\) obtained by attaching copies of Q to the vertices of \(X_ N\), the maximal elements of P. If P satisfies (A.1) and (A.2), then \(d(x,y)=N-rank(x\wedge y)\) is a metric on \(X_ N\times X_ N.\)

Letting \(f^{\alpha}(\beta)=1\) if \(\alpha\leq \beta\), 0 otherwise, then \(\{f^{\alpha}:\) \(\alpha\in P\}\) is a basis for \(L^ 2(P)\). If Hom(n) is the collection of complex valued functions defined on \(P=span\{f^{\alpha}:\) \(\alpha \in X_ n\}\) and if \(Harm(n)=Hom(n)\wedge Ker d\), where \(df^{\alpha}=\sum_{\beta}f^{\beta}\), \(\alpha\) covers \(\beta\), then conditions (A.1)-(A.4) are sufficient to insure that the restrictions of the functions in Harm(n) to \(X_ N\) yield irreducible G-modules, and that the representation of G on \(L^ 2(X_ N)\) is multiplicity free with the decomposition \(L^ 2(X_ N)=\oplus^{N}_{n=0}Harm(n)|_{X_ N}\). Moreover, the normalized ”spherical” function for \(Harm(n)|_{X_ N}\) is \(f_ n(\Omega_{nj})=\sum^{n}_{i=0}c_ iF_{ni}(\Omega_{Nj})/d_ n\), where \(F_{ni}=\sum_{\alpha \in \Omega_{ni}}f^{\alpha}\) and \(\{F_{ni}:\) \(0\leq i\leq n\}\) is a basis for the right H-invariant functions in Hom(n). Also \(dF_{ni}=a_{n-1,i}F_{n-1,i}+b_{n- 1,i}F_{n-1,i-1}\), where \(a_{n-1,i}=| \{\gamma \in \Omega_{ni}:\) \(\gamma >\alpha \}|\), \(b_{n-1,i}=| \gamma \in \Omega_{n,i+1}:\) \(\gamma >\alpha \}| \neq 0\) for some \(a\in \Omega_{n-1,j}\), \((-1)^ ic_ i(\prod^{i-1}_{j=0}a_{n- 1,j}/b_{n-1,j})c_ 0\), and \(d_ n=| \{\alpha \in X_ n:\) \(\alpha \leq x^{(0)}\}|\).

Reviewer: J.Neggers (Tuscaloosa)

##### MSC:

06A06 | Partial orders, general |

33C55 | Spherical harmonics |

43A15 | \(L^p\)-spaces and other function spaces on groups, semigroups, etc. |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

##### Keywords:

finite poset; maximal elements; complex valued functions; automorphism group; permutation representation; irreducible G-modules; principal harmonics; orthogonal polynomials; ranked poset
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\textit{D. Stanton}, J. Comb. Theory, Ser. A 40, 136--149 (1985; Zbl 0573.06001)

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