A.R. Calderbank, P. Hanlon and R.W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. 53 (1986), 288--320.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to extend this method to other non supersolvable lattices such as those for the exceptional roots systems, lattices associated with certain free arrangements of hyperplanes [20, 11, 12] the generalized Dowling lattices [8] certain lattices associated with partitions into even and odd block size [5], and frame matroid lattices [23] Since some of these posets are not semimodular lattices, an appropriate analog of Theorem 1.1 will also have to be found for such cases. 2) The reader may be wondering how to construct an ADT, T , for a given lattice, L. The following method, while lengthy, may ....

A. R. Calderbank, P. Hanlon and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (1986), 288--320.

.... with rank selection, and, using the Hopf trace formula, derived formulas for the trace of an automorphism on the rank selected homology ( see [St1] Note that the rank selected subposet Pi 2n considered in this paper is different from the 2 divisible sublattice of Pi 2n studied in [CHR] and [W2] It is well known that for Cohen Macaulay posets, or more generally for posets with a unique nonvanishing homology space, the dimension of the homology space is the absolute value of the Mobius number of the poset (see [St2] For Pi 2n ; the dimension of the homology module can be ....

A. R. Calderbank, P. Hanlon, and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. 53 (1986), no. 3, 288-320.

....on the character values of the representation of G on the homology of P is illustrated with the example of Sn acting on the Boolean lattice of subsets of an n element set. For Cohen Macaulay posets, this approach was first pursued by Stanley ( St1] and later by Calderbank, Hanlon and Robinson ([CHR]) In Section 2 we focus on the action of the symmetric group, by permutation of coordinates, on products of posets. We show how, under certain conditions, the Hopf trace formula can be used to give a new, simple proof of the existence of sign twists in the action of Sn on the homology of a ....

....Another product, called the reduced product, is introduced and discussed in Section 2. The reduced product is a proper subposet of the ordinary product of posets (with unique minimal and maximal elements) and is exemplified by lower intervals in the d divisible partition lattice (see [Bj3] [CHR] and [Wa] The corresponding d divisible orbit arrangement of [Bj3] is thus one example of a subspace arrangement in which reduced products arise. The homology representation for lower intervals in this case was determined in [Su] and applied to the cohomology of the arrangement and its orbit ....

[Article contains additional citation context not shown here]

A. R. Calderbank, P. Hanlon, and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. 53 (1986), no. 3, 288-320.

....for a general class of subposets of the partition lattice induced by restricting the block sizes of the partitions. This general class contains subposets which are nonpure shellable. Homology of restricted block size partition posets was first considered by Calderbank, Hanlon and Robinson [CHR] who derived beautiful plethystic formulas yielding the character of the representation of the symmetric group on the top homology of the d divisible partition lattice (all block sizes divisible by some fixed d) and the 1 mod d partition lattice (all block sizes congruent to 1 mod d) both of ....

....block sizes divisible by some fixed d) and the 1 mod d partition lattice (all block sizes congruent to 1 mod d) both of which are pure. A key fact used in the Calderbank, Hanlon and Robinson proof is that the two posets are pure shellable which was proved respectively by Wachs and Bjorner (cf. [CHR] and [Sa] For general k, the k mod d partition poset is not pure and that is where the difficulty lies in computing the homology representations of the general k mod d partition posets. In [Su2] Sundaram gives a formula for the virtual representation of the symmetric group on the alternating ....

[Article contains additional citation context not shown here]

A.R. Calderbank, P. Hanlon, and R.W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (3) 53 (1986), 288--320.

....result of Stanley [St1] the tangent number E 2n Gamma1 is itself the Betti number of a subposet of the partition lattice, namely, the join sublattice Pi n of Pi 2n generated by the partitions with n blocks all of size 2. The homology representation of S 2n for this lattice was determined in [CHR]. v) Finally, the Euler number Em Gamma1 is also the Betti number of the subposet of the Boolean lattice Bm consisting of alternate ranks. The homology representation of Sm in this case was computed by Solomon ( So] to be one of the Foulkes representations: it is the representation of Sm indexed ....

....: In particular fi 2n = fi 2n;2 : As in the proof of Proposition 1. 2, we obtain the following generating function for the Betti numbers fi m;d : 1 Gamma e ) d Gamma1 fi nd i;d nd i (nd i) fi nd;d (nd) Denote by Pi n the d divisible lattice first studied in [CHR], that is, the sublattice of Pi nd consisting of partitions whose block sizes are divisible by d: Remark 1.4.1 (iv) points out the relation between the Betti numbers of Pi 2n = P 2n;2 and Pi n : We do not know if there is any relation between the Betti number fi nd;d of the poset P nd;d ....

A. R. Calderbank, P. Hanlon and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (3), 53 (1986), 288-320.

....determines the homology representation on the subposet Pi n of partitions of n elements into blocks of size congruent to 1 mod 2: The idea behind our basic observation, Lemma 1. 1, is implicitly used to calculate the Mobius functions of fixed point posets by Calderbank, Hanlon and Robinson ([CHR], Theorem 2.2) Our approach has the advantage of being more conceptual from the representation theoretic point of view: we avoid intricate Mobius function computations by manipulating virtual homology modules, and by exploiting the considerable machinery of symmetric function theory as presented ....

.... The rank one elements in the poset are the (nd) n d n partitions all of whose blocks have size d: In general, the rank of a partition x is given by n 1 minus the number of blocks of x: The rank of Pi n is thus n: The homology representation of S nd on Pi n was first studied in [CHR]. We denote the Frobenius characteristic for this representation by n : Corollary 4.7 of [CHR] is essentially a generating function for the n s. We show how to derive this using our methods. As in (i) we compute the Whitney homology of the dual of Pi n : Again the salient factor is that ....

[Article contains additional citation context not shown here]

A. R. Calderbank, P. Hanlon and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (3), 53 (1986), 288-320.

....were introduced in [Bj4] as the intersection lattices of Bjorner s orbit arrangements. The k divisible lattice Pi (k ) is the intersection lattice of the orbit arrangement corresponding to the partition (k ) of n ( Bj4, Example 3. 3] This lattice has been studied by various authors ( C H R] Sa] Wa] In particular it follows from [Sa] and [Wa] that Delta( Pi (k ) is a Cohen Macaulay space, and the Betti numbers are known ( St1] We carry out (STEP 3) for the manifolds M for = k; 1 For a given poset P our reduction (STEP 1) transforms all problems into problems ....

....WH 0 ( Pi (k;1 ) is the trivial Sn module. Finally we record the same calculations for the lattice Pi (k ) consisting of partitions in Pi k all of whose block sizes are divisible by k: The representation of the symmetric group on the top homology of Pi (k ) was first determined in [C H R] For the purposes of the computations in the next section, we shall need the representation on the Whitney homology as described in [Su] Denote the characteristic of the action of S kl on the (top) homology e H Gamma2 ( Pi (k ) by : We have Theorem (3.2) Su, Lemma 1.1 and ....

[Article contains additional citation context not shown here]

Calderbank, A., Hanlon, P., Robinson, R.: Partitions into even and odd block sizes and some unusual characters of the symmetric groups. Proc. London Math. Soc. 53(3), 288--320 (1986)

....result of Stanley [St1] the tangent number E 2n Gamma1 is itself the Betti number of a subposet of the partition lattice, namely, the join sublattice Pi 2 n of Pi 2n generated by the partitions with n blocks all of size 2. The homology representation of S 2n for this lattice was determined in [CHR]. v) Finally, the Euler number Em Gamma1 is also the Betti number of the subposet of the Boolean lattice Bm consisting of alternate ranks. The homology representation of Sm in this case was computed by Solomon ( So] to be one of the Foulkes representations: it is the representation of Sm ....

....the following generating function for the Betti numbers fi m;d : Proposition 4. 1 (1 Gamma e x ) d Gamma1 X i=1 X n0 ( Gamma1) n 1 fi nd i;d x nd i (nd i) X n1 ( Gamma1) n fi nd;d (e x Gamma 1) nd (nd) Denote by Pi d n the d divisible lattice first studied in [CHR], that is, the sublattice of Pi nd consisting of partitions whose block sizes are divisible by d: Remark 1.4.1 (iv) points out the relation between the Betti numbers of Pi e 2n = P 2n;2 and Pi 2 n : We do not know if there is any relation between the Betti number fi nd;d of the poset P nd;d ....

A. R. Calderbank, P. Hanlon and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (3), 53 (1986), 288-320.

....on the character values of the representation of G on the homology of P is illustrated with the example of Sn acting on the Boolean lattice of subsets of an n element set. For Cohen Macaulay posets, this approach was first pursued by Stanley ( St1] and later by Calderbank, Hanlon and Robinson ([CHR]) In Section 2 we focus on the action of the symmetric group, by permutation of coordinates, on products of posets. We show how, under certain conditions, the Hopf trace formula can be used to give a new, simple proof of the existence of sign twists in the action of Sn on the homology of a ....

....Another product, called the reduced product, is introduced and discussed in Section 2. The reduced product is a proper subposet of the ordinary product of posets (with unique minimal and maximal elements) and is exemplified by lower intervals in the d divisible partition lattice (see [Bj3] [CHR] and [Wa] The corresponding d divisible orbit arrangement of [Bj3] is thus one example of a subspace arrangement in which reduced products arise. The homology representation for lower intervals in this case was determined in [Su] and applied to the cohomology of the arrangement and its orbit ....

[Article contains additional citation context not shown here]

A. R. Calderbank, P. Hanlon, and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. 53 (1986), no. 3, 288-320.

....the homology representation on the subposet Pi (1;2) n of partitions of n elements into blocks of size congruent to 1 mod 2: The idea behind our basic observation, Lemma 1. 1, is implicitly used to calculate the Mobius functions of fixed point posets by Calderbank, Hanlon and Robinson ([CHR], Theorem 2.2) Our approach has the advantage of being more conceptual from the representation theoretic point of view: we avoid intricate Mobius function computations by manipulating virtual homology modules, and by exploiting the considerable machinery of symmetric function theory as presented ....

.... The rank one elements in the poset are the (nd) n d n partitions all of whose blocks have size d: In general, the rank of a partition x is given by n 1 minus the number of blocks of x: The rank of Pi d n is thus n: The homology representation of S nd on Pi d n was first studied in [CHR]. We denote the Frobenius characteristic for this representation by d n : Corollary 4.7 of [CHR] is essentially a generating function for the d n s. We show how to derive this using our methods. As in (i) we compute the Whitney homology of the dual of Pi d n : Again the salient factor is ....

[Article contains additional citation context not shown here]

A. R. Calderbank, P. Hanlon and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (3), 53 (1986), 288-320.

.... with rank selection, and, using the Hopf trace formula, derived formulas for the trace of an automorphism on the rank selected homology ( see [St1] Note that the rank selected subposet Pi e 2n considered in this paper is different from the 2 divisible sublattice of Pi 2n studied in [CHR] and [W2] It is well known that for Cohen Macaulay posets, or more generally for posets with a unique nonvanishing homology space, the dimension of the homology space is the absolute value of the Mobius number of the poset (see [St2] For Pi e 2n ; the dimension of the homology module can be ....

A. R. Calderbank, P. Hanlon, and R. W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. 53 (1986), no. 3, 288-320.

....homology for a general class of subposets of the partition lattice induced by restricting the block sizes of the partitions. This general class contains subposets which are nonpure shellable. Homology of restricted block size partition posets was first considered by Calderbank, Hanlon and Robinson [CHR] who derived beautiful plethystic formulas yielding the character of the representation of the symmetric group on the top homology of the d divisible partition lattice (all block sizes divisible by some fixed d) and the 1 mod d partition lattice (all block sizes congruent to 1 mod d) both of ....

....block sizes divisible by some fixed d) and the 1 mod d partition lattice (all block sizes congruent to 1 mod d) both of which are pure. A key fact used in the Calderbank, Hanlon and Robinson proof is that the two posets are pure shellable which was proved respectively by Wachs and Bjorner (cf. [CHR] and [Sa] For general k, the k mod d partition poset is not pure and that is where the difficulty lies in computing the homology representations of the general k mod d partition posets. In [Su2] Sundaram gives a formula for the virtual representation of the symmetric group on the alternating ....

[Article contains additional citation context not shown here]

A.R. Calderbank, P. Hanlon, and R.W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. (3) 53 (1986), 288--320.

No context found.

A.R. Calderbank, P. Hanlon and R.W. Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc. London Math. Soc. 53 (1986), 288--320.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC