Explicit expression for the product of the classes of fixed-point free involutions

When mathrm {Sp}(2n,mathbb {C}) acts on the flag variety of mathrm {SL}(2n,mathbb {C}), the orbits are in bijection with fixed point free involutions in the symmetric group S_{2n}. In this case, the associated Kazhdan–Lusztig–Vogan polynomials P_{v,u} can be indexed by pairs of fixed point free involutions vge u, where ge denotes the Bruhat order on S_{2n}. We prove that these polynomials are combinatorial invariants in the sense that if f:[u,w_0]rightarrow [u',w_0] is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then P_{v,u} = P_{f(v),u'} for all vge u.

was defined to be the least integer n for which there is an equivariant map X -+s n. We abbreviate this invariant to co-ind X. In this terminology the classical Borsuk theorem states that co-ind Sn = n. There are also numerous results (for references, see [2]) which among other things relate co-index to the homology of the quotient space X/T. The main purpose of the present note is the computation of the coindex in several examples in which homotopy, rather than homology, considerations are of primary importance. It should be mentioned that A. S. Svarc has also recently studied the application of homotopy theory to equivariant maps [5]; there is a considerable overlap between his work and our previous paper [2]. We consider as in our previous paper the space p(Sn) of paths on Sn which join a given point x to its antipode A(x) = -x together with the natural involution of p(Sn). It is shown that co-ind P(Sn) = n for n : 1, 2, 4 or 8. Next we consider the space V(Sn) of unit tangent vectors to sn, with its involution (the antipodal map on each fibre), and show that co-ind V(Sn) = n for n : 1, 3, or 7 and co-ind V(Sn) = n - 1 for n = 1, 3 or 7. We also compute the co-index of involutions on low dimensional projective spaces. The arguments rely on suspension and Hopf invariant theorems, using particularly the results of J. F. Adams [1] on maps of Hopf invariant one. 2. The space of paths P(Sn). We choose a base point x e Sn and we let P(Sn) denote the space of all paths in Sn which join x to its antipode - x. A fixed point free involution on P(Sn) is given by 17(p)(t) =-p(l - t), where p(t) is a point in P(S). In this section we show

We determine the possible $\mathbb{Z}_{2}$-cohomology rings of orbit spaces of free actions of $\mathbb{Z}_{2}$ (or fixed point free involutions) on the Dold manifold $P(1,n)$, where $n$ is an odd natural number.

Let β be a given permutation of [n]={1,2,...,n} of type (β1, β2,...,βn) (i.e., β has β1 cycles of length i; ∑iβ1 = n. We find (in terms of the β1's and bijectively) the number of endofunctions, permutations, cyclic permutations, derangements, fixed point free involutions, assemblies of octopuses, octopuses, idempotent endofunctions, rooted trees (i.e. contractions), forests of rooted trees, trees, vertebrates, relations (digraphs), symmetric relations (simple graphs), partitions, and connected endofunctions on [n], kept fixed by the natural action (byconjugation) of β. This approach leads to algorithms generating these structures.

1. Introduction.We shall continue our discussion of fixed point free involutions which was begun in [2].We denote by S" the antipodal involution on the n-sphere.For any fixed point free involution on a space X the co-index was defined to be the least integer n for which there is an equivariant map X -* S".We abbreviate this invariant to co-ind X.In this terminology the classical Borsuk theorem states that co-ind S" = n.There are also numerous results (for references, see [2]) which among other things relate co-index to the homology of the quotient space X/T.The main purpose of the present note is the computation of the coindex in several examples in which homotopy, rather than homology, considerations are of primary importance.It should be mentioned that A. S. Svarc has also recently studied the application of homotopy theory to equivariant maps [5]; there is a considerable overlap between his work and our previous paper [2].We consider as in our previous paper the space P(S") of paths on S" which join a given point x to its antipode A(x) = -x together with the natural involution of P(S").It is shown that co-ind P(S") = n for n ^ 1, 2, 4 or 8. Next we consider the space V(S") of unit tangent vectors to S", with its involution (the antipodal map on each fibre), and show that co-ind V(S") = n for n # 1, 3, or 7 and co-ind V(S") = n -1 for n = 1, 3 or 7. We also compute the co-index of involutions on low dimensional projective spaces.The arguments rely on suspension and Hopf invariant theorems, using particularly the results of J. F. Adams [1] on maps of Hopf invariant one.2. The space of paths P(S").We choose a base point xeS" and we let P(S") denote the space of all paths in S" which join x to its antipode -x.A fixed point free involution on P(S") is given by T(p)(t) = -p(l -t), where p(t) is a point in P(S").In this section we show (2.1) Theorem.For n # 1, 2, 4 or 8, co-ind P(S") = n.We showed this for n > 1 and odd in [2, p. 425] and we conjectured this result as the general case.We see first that co-ind P(S") = n by defining an equivariant map m : P(S") -► S" as m(p(t)) = p(l/2) e S".Now we suppose there is an equivariant map mx : P(S") -> Sn_1.We define an

In this paper, we present a geometric construction of the Moufang quadrangles discovered by Richard Weiss (see Tits & Weiss [18] or Van Maldeghem [19]). The construction uses fixed point free involutions in certain mixed quadrangles, which are then extended to involutions of certain buildings of type F4. The fixed flags of each such involution constitute a generalized quadrangle. This way, not only the new exceptional quadrangles can be constructed, but also some special type of mixed quadrangles.

The space P(Sn) of all paths a) in Sn with given initial point x and endpoint -x admits an involution (To)(t) = -co(1 t). With the standard antipodal involution on Sn-1 an equivariant map P(Sn) -* Sn-' is constructed for n = 2, 4, or 8. Afixedpointfree involution on a space Xis a map T:X T Xsatisfying T' = identity and Tx $ x for every x E X. Three examples of interest are (i) (Sn, T1) with T1x = -x; (ii) (V(Sn), T2) where V(Xn) = the unit tangent sphere bundle of Sn and T2 = the antipodal action on each fibre; and (iii) (P(Sn), T3) where P(Sn) = the space of paths with given initial point x E Sn and endpoint -x, and (T3w))(t) = -w(1 t). For (X, T) a fixed point free involution the co-index of X is the least integer n for which there exists an equivariant map (X, T) --(Sn, T1). A classical result of Borsuk [1] asserts that the co-index of (Sn, T1) equals n. In [2] Conner and Floyd determined the co-index of (V(Sn), T2) (for all n) and the co-index of (P(Sn), T3) for all n except n = 2, 4, and 8. Their results assert (i) co-index(V(Sn), T2) = n or n 1 according as n 0 {1, 3, 7} or n E {1, 3, 7} and (ii) co-index(P(Sn), T3) = n if n 0 {1, 2, 4, 8} and = n 1 if n = 1. The remaining cases of (ii) are resolved by PROPOSITION. For n = 2, 4, or 8 there exists an equivariant map (P(Sn), T3) -_ (Sn-i, T1). PROOF. For n = 2, 4, and 8 there are the Hopf fibrations Sn-1 F S2n-1 P+ Sn. Here S2n-1 is the unit sphere in F2 (F = complexes for n = 2, the quaternions for n = 4 and the Cayley numbers for n = 8) and the map p assigns to each unit vector the 1-dimensional (over F) subspace it spans. Fix a point x E Sn and a point y E S7n= the fibre of p over x. S2n-1 iS the join of Sxnand Snx71, where Sf7xl is both the fibre over -x and the unit sphere in the 1-dimensional (over F) subspace orthogonal to the subspace spanned by Sn-1. Moreover p maps the great circle arc Received by the editors December 8, 1970. AMS 1969 subject classifications. Primary 5536.

The space $P({S^n})$ of all paths $\omega$ in ${S^n}$ with given initial point $x$ and endpoint $- x$ admits an involution $(T\omega )(t) = - \omega (1 - t)$. With the standard antipodal involution on ${S^{n - 1}}$ an equivariant map $P({S^n}) \to {S^{n - 1}}$ is constructed for $n = 2,4,$, or $8$.

A transfer spectral sequence for fixed point free involutions with an application to stunted real projective spaces

We give an explicit construction of infinite sharply 2-transitive groups with fixed point free involutions and without nontrivial abelian normal subgroup.

The distribution of descents in certain conjugacy classes of $$S_n$$ has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings.

In [1], P. A. Smith has proved that the set of fixed points of a periodic homeomorphism T of the 3-sphere S3 onto itself is an i-sphere imbedded in S3, i = -1, 0, 1, or 2, where the -1-sphere denotes the empty set. In trying to prove that an involution (map of period 2) on S3 is conjugate, in the group of all homeomorphisms of S3 onto itself, to an orthogonal involution, it is natural to divide the problem into four cases, as the dimension, n, of the fixed point sphere is -1, 0, 1, or 2, respectively. R. H. Bing, [2], has settled the case n=2 in the negative, unless one assumes that the 2-sphere of fixed points is tamely imbedded. Regarding the case n= 1, Montgomery and Samelson, [3], have shown that, if T is semi-linear, the circle of fixed points cannot be a knotted torus knot of type (p, 2), and moreover, that if the circle of fixed points is not knotted, then the involution, T, is conjugate to an orthogonal involution. This latter result suggests an approach for the case n = -1. An invariant set will mean a set mapped onto itself by the involution. In the absence of fixed points, the invariant circles become important. In fact, we have:

Fixed Point Free Involutions and Equivariant Maps. II

An involution is a permutation that is its own inverse. Given a permutation [sigma] of [n], let N_n([sigma]) denote the number of ways to write [sigma] as a product of two involutions. The random variables N_n are asymptotically lognormal when the symmetric groups S_n are equipped with uniform probability measures, in particular, and more generally, Ewens measures of some fixed parameter [theta] > 0. The proof is based upon the observation that, for most permutations [sigma], the number of involution factorizations N_n([sigma]) can be well-approximated by B_n([sigma]), the product of the cycle lengths of [sigma]. The asymptotic lognormality of N_n can therefore be deduced from Erdős and Turán's theorem that B_n is itself asymptotically lognormal. We then briefly consider fixed-point free involution factorizations. A necessary and sufficient condition for a permutation to be the composition of two fixed-point free involutions is for it to have an even number of k-cycles, k = 1, 2, ... Through a combination of singularity analysis, the method of moments, and an appeal to the Shepp-Lloyd model for random permutations, the asymptotic enumeration and cycle structure of random permutations admitting fixed-point free involution factorizations are calculated.

The Eulerian distribution on involutions is indeed unimodal