S. Sundaram, The homology representations of the symmetric group on Cohen--Macaulay subposets of the partition lattice, Adv. in Math. 104 (1994), 225--296.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....=x for 1 for n 4. What can be said about the ab index of the complex D n arrangement where # is a kth primitive root of unity The homology of the partition lattice # n 1 L n,1 has been extensively studied in order to find representations of the symmetric group; see [23] and [28]. In the same spirit, Wachs has studied the signed partition lattice L n,2 ; see [29] What can be said about the representations of the symmetric group arising from the Dowling lattice L n,k A related question is to find an explicit basis for the highest homology group of the Dowling lattice, ....

S. Sundaram, The homology representations of the symmetric group on Cohen--Macaulay subposets of the partition lattice, Adv. in Math. 104 (1994), 225--296.

..... There is a relation between the cd index of the Boolean algebra B n and certain classes of permutations. For instance, the cd index of B n is a refined enumeration of Andre permutations [17] Similarly, it is also a refined enumeration of simsun permutations, first defined by Simion and Sundaram [22, 23]. Another known example of a poset permutations pair is the cubical lattice and signed Andre permutations [8, 17] This motivates us to ask the following question. Given an Eulerian poset P , is it possible to find a canonical class of permutations which correspond to the cd index of the poset P ....

S. Sundaram, The Homology Representation of the Symmetric Group on Cohen-Macaulay Subposets of the Partition Lattice, Adv. Math. 104 (1994), 225-296.

....homology of P to be If a finite group G acts as a group of automorphisms of P then we say that P is a G poset. This action induces a representation of G on each homology and Whitney homology of P . The following important relationship between these G modules was established by Sundaram [Su1] as a consequence of the Hopf trace formula. Proposition 1.1. Sundaram [Su1] Let P be a G poset. Then the following isomorphism of virtual sums of G modules holds: 1 1) H r Gamma2 (P ) Corollary 1.2. Sundaram [Su1] Let P be a G poset of length 1. If H i (P ) vanishes ....

....of P then we say that P is a G poset. This action induces a representation of G on each homology and Whitney homology of P . The following important relationship between these G modules was established by Sundaram [Su1] as a consequence of the Hopf trace formula. Proposition 1.1. Sundaram [Su1]) Let P be a G poset. Then the following isomorphism of virtual sums of G modules holds: 1 1) H r Gamma2 (P ) Corollary 1.2. Sundaram [Su1] Let P be a G poset of length 1. If H i (P ) vanishes for all i 6= Gamma 2 then as a virtual sum of G modules one has, 1 2) H ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Advances in Math. 104 (1994), 225--296.

....[6, Theorem 1] or [5, page 120, Theorem 2.4] This allows us in Corollary 4.2 to translate (a weaker version of) our result into the language of the representation theory. Questions of such nature have been studied before, most notably by R. Stanley in [13] P. Hanlon in [10] and S. Sundaram in [14]. We give more conceptual proofs for [13, Prop. 7.8] 10, Theorem 3.1] and [14, Theorem 4.2] as the special cases of our result (the authors were working with the case when is coming from a rank selection and k = 2) We hope that this sheds some light on the role which the number partitions of ....

....also follows from Theorem 4.1 that if is a rank selection such that the ranks 1; b n 2 c are among the selected ones, then the multiplicity of the trivial character as above is 0. Up to a few border cases, this is [14, Theorem 4.2] We 6 DMITRY N. KOZLOV refer the interested reader to [14] for a thorough account of known results and open questions about the multiplicity of the trivial character in the representation of Sn on the homology groups of rank selections of the partition lattice, as well as for an extensive amount of computational data. In general, in our terminology, all ....

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104, (1994), pp. 225-296.

....1g H r Gamma2 ( 0; x) If a finite group G acts as a group of automorphisms of P then we say that P is a G poset. This action induces a representation of G on each homology and Whitney homology of P . The following important relationship between these G modules was established by Sundaram [Su1] as a consequence of the Hopf trace formula. Proposition 1.1. Sundaram [Su1] Let P be a G poset. Then the following isomorphism of virtual sums of G modules holds: 1 1) M r=1 ( Gamma1) r H r Gamma2 (P ) Gamma1 M r=0 ( Gamma1) r 1 WH r (P ) Corollary 1.2. Sundaram [Su1] ....

....of P then we say that P is a G poset. This action induces a representation of G on each homology and Whitney homology of P . The following important relationship between these G modules was established by Sundaram [Su1] as a consequence of the Hopf trace formula. Proposition 1.1. Sundaram [Su1]) Let P be a G poset. Then the following isomorphism of virtual sums of G modules holds: 1 1) M r=1 ( Gamma1) r H r Gamma2 (P ) Gamma1 M r=0 ( Gamma1) r 1 WH r (P ) Corollary 1.2. Sundaram [Su1] Let P be a G poset of length 1. If H i (P ) vanishes for all i 6= ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Advances in Math. 104 (1994), 225--296.

....of more familiar posets. This leads to a decomposition for the Sn module structure of the homology, which is presented in Theorem 2.8. The theory of symmetric functions ( Macd] allows us to carry out representationtheoretic calculations in a compact manner. Using the techniques developed in [Su1] for determining representations of Sn on the homology of subposets of the partition lattice, in Section 3 we determine the Sn action on each graded piece of the homology of Pi (k;1 n Gammak ) Theorem 3.5) The Frobenius characteristics of these modules have an elegant expression as a plethysm ....

....for both a (virtual) representation of Sn and its Frobenius characteristic. In particular, we use the notation hn , e n and s for both the corresponding irreducible representation and its Frobenius characteristic (i.e. the Schur function for this shape) We need some notation and results of [Su1, Section 1], which we review here briefly. Let be an integer partition of n with t parts and let m i denote the multiplicity of the part i in . For any (possibly virtual) representation fl of S t ; and any sequence of (possibly negative) representations R i of S i ; denote by the representation ....

[Article contains additional citation context not shown here]

Sundaram, S.: The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Advances in Math. 104 (No.2) (March 1994), 225-296.

....2n ( k) k = 1; n Gamma 1; are all polynomials in the homogeneous symmetric functions fh i : i 1g with nonnegative integer coefficients. An interesting combinatorial invariant of these homology representations is the multiplicity of the trivial representation (see [St1, Section 7] and [Su1, Sections 3 and 4]) If the preceding conjectures were true, the multiplicity fi 2n;k of the trivial S 2n module in the homology of Pi k) would simply be the number of S 2n orbits. Note that from Theorem 2.1 we have fi 2n;n = b i (n) In particular the sum i=2 b i (n) is known to be nonnegative. The ....

....In particular, we have the formula Corollary 4.10. For symmetric functions f and g; h 2 [f g] h 2 [f ] h 2 [g] f Delta g: The interested reader can now recover (2.2) from the computation of (4.2) We record one more example of a plethystic computation which is frequently useful. See [Su1] for applications to computing the multiplicity of the trivial representation in homology modules. These results are easily derived from the properties developed in this section. Proposition 4.11. Let be an integer partition of n: Then (i) s [1 q] q r 1 Gammaq n Gammar 1 1 Gammaq ; ....

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2) (1994), 225-296.

....representation on the cohomology of the complement of a subspace arrangement to the problem of computing the homology representation on the intersection lattice of the arrangement. Other methods for computing homology representations of posets arising from the partition lattice were introduced in [Su], and were applied to the k equal partition lattice ( SWa] and the corresponding subspace arrangement ( SWe1] Throughout the paper we will assume the reader is familiar with the ordinary representation theory of finite groups. The first two sections of this paper are written with the ....

....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 space in [SWe1] A more general class of subposets of the partition lattice whose lower intervals admit a reduced product structure, is discussed in Section 4. The discussion in Section 2 leads to a general theorem (Theorem 2.10) on ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2) (1994), 225-296.

....identity of Stanley (see, e.g. S2, Example 3.6] 5.1) 4 5 = h 1 : Let A n;k denote the degree n term in 5 : The identity (5.1) may thus be rewritten A n;1 = h 1 : We also deduce the formulas (5. 2) A n;k = 0; k n 2; A k 1;k = Gamma( Gamma1) k 1 Recall from [S1] that if = m i n denotes an integer partition of n with m i parts of size i; then the degree n term in the left hand side of (5.1) is obtained by computing the following sum of products of plethysms: 5.3) n i m i Y hm i [ Gamma1) i ] The key observation now is that, ....

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2)(1994), 225-296.

....of P 4 and the 1 dimensional order complex of Q 4 both have the same homotopy type, and hence have the same homology. Figure 1: The poset b P 4 ffi ffi ffi ffi ffi ffi 12=3=4 34=1=2 13=2=4 24=1=3 14=2=3 23=1=4 Figure 2: The poset b Q 4 We describe briefly the motivation for this work. In [Su] some general techniques were developed for computing the homology representation of a poset for a finite group of automorphisms, and applied to Cohen Macaulay subposets of the partition lattice. Note that the subposets P n are invariant under the action of the symmetric group Sn : In ....

....4) 3. The representation of the symmetric group Sn on the homology In this section all homology is taken over the field of complex numbers. We shall first compute the Sn module structure of the unique nonvanishing homology of the poset Q n : For this we need to recall some of the results of [Su]. For a finite poset Q and a finite group G of autormorphisms of Q; we denote by Alt(Q) the Lefschetz (G )module of Q; i.e. Alt(Q) i H i (Q) Theorem 3.1. See [Su, Theorem 1.10 and Remark 1.11] Let P be a CohenMacaulay poset of rank r; G a finite group of automorphisms of P; and Q a ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2)(1994), 225-296. 17

....2n : We show that the homology of Pi 2n has dimension 2 2n Gamma1 E 2n Gamma1 ; where E 2n Gamma1 is the tangent number. It is thus an integral multiple of both the Genocchi number and an Andr e or simsun number. Using the general theory of rank selected homology representations developed in [Su], we show that, for the special case of Pi 2n ; the character of the symmetric group S 2n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers b i (n) 2 i n; defined recursively. We conjecture that, for the full ....

....subposet Pi n is also. In general, Pi n is not a lattice. In this paper we study the representation of the symmetric group Sn on the homology of the subposet Pi A systematic study of the homology representations of rank selected and other Cohen Macaulay subposets of Pi n was initiated in [Su], where a number of tools were introduced. We shall use these techniques to show that, when n is even, the homology representation of Pi n has remarkable properties. The formulas in [Su, Theorem 2.8] can be programmed using Stembridge s symmetric functions package for Maple, and first allowed us ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2)(1994), 225-296.

....of P and the 1 dimensional order complex of Q 4 both have the same homotopy type, and hence have the same homology. Figure 1: The poset b P 4 ffi ffi ffi ffi ffi ffi 12=3=4 34=1=2 13=2=4 24=1=3 14=2=3 23=1=4 Figure 2: The poset b Q 4 We describe briefly the motivation for this work. In [Su1] some general techniques were developed for computing the homology representation of a poset for a finite group of automorphisms, and applied to Cohen Macaulay subposets of the partition lattice. Note that the subposets P n are invariant under the action of the symmetric group Sn : In ....

....4) 3. The representation of the symmetric group Sn on the homology In this section all homology is taken over the field of complex numbers. We shall first compute the Sn module structure of the unique nonvanishing homology of the poset Q n : For this we need to recall some of the results of [Su1]. For a finite poset Q and a finite group G of automorphisms of Q; we denote by Alt(Q) the Lefschetz (G )module of Q; i.e. Alt(Q) i H i (Q) Theorem 3.1. See [Su1, Theorem 1.10 and Remark 1.10.1] Let P be a CohenMacaulay poset of rank r; G a finite group of automorphisms of P; and Q ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2)(1994), 225-296.

....show that the homology of Pi e 2n has dimension (2n) 2 2n Gamma1 E 2n Gamma1 ; where E 2n Gamma1 is the tangent number. It is thus an integral multiple of both the Genocchi number and an Andr e or simsun number. Using the general theory of rank selected homology representations developed in [Su], we show that, for the special case of Pi e 2n ; the character of the symmetric group S 2n on the homology is supported on the set of involutions. Our proof techniques lead to the discovery of a family of integers b i (n) 2 i n; defined recursively. We conjecture that, for the full ....

.... Pi e n is also. In general, Pi e n is not a lattice. In this paper we study the representation of the symmetric group Sn on the homology of the subposet Pi e n : A systematic study of the homology representations of rank selected and other Cohen Macaulay subposets of Pi n was initiated in [Su], where a number of tools were introduced. We shall use these techniques to show that, when n is even, the homology representation of Pi e n has remarkable properties. The formulas in [Su, Theorem 2.8] can be programmed using Stembridge s symmetric functions package for Maple, and first allowed ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2)(1994), 225-296.

....representation on the cohomology of the complement of a subspace arrangement to the problem of computing the homology representation on the intersection lattice of the arrangement. Other methods for computing homology representations of posets arising from the partition lattice were introduced in [Su], and were applied to the k equal partition lattice ( SWa] and the corresponding subspace arrangement ( SWe1] Throughout the paper we will assume the reader is familiar with the ordinary representation theory of finite groups. The first two sections of this paper are written with the ....

....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 space in [SWe1] A more general class of subposets of the partition lattice whose lower intervals admit a reduced product structure, is discussed in Section 4. The discussion in Section 2 leads to a general theorem (Theorem 2.10) on ....

[Article contains additional citation context not shown here]

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2) (1994), 225-296.

....of more familiar posets. This leads to a decomposition for the Sn module structure of the homology, which is presented in Theorem 2.8. The theory of symmetric functions ( Macd] allows us to carry out representationtheoretic calculations in a compact manner. Using the techniques developed in [Su1] for determining representations of Sn on the homology of subposets of the partition lattice, in Section 3 we determine the Sn action on each graded piece of the homology of Pi (k;1 n Gammak ) Theorem 3.5) The Frobenius characteristics of these modules have an elegant expression as a plethysm ....

....for both a (virtual) representation of Sn and its Frobenius characteristic. In particular, we use the notation hn , e n and s for both the corresponding irreducible representation and its Frobenius characteristic (i.e. the Schur function for this shape) We need some notation and results of [Su1, Section 1], which we review here briefly. Let be an integer partition of n with t parts and let m i denote the multiplicity of the part i in . For any (possibly virtual) representation fl of S t ; and any sequence of (possibly negative) representations R i of S i ; denote by Res Theta j Sm j fl ....

[Article contains additional citation context not shown here]

Sundaram, S.: The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Advances in Math. 104 (No.2) (March 1994), 225-296.

....2n ( k) k = 1; n Gamma 1; are all polynomials in the homogeneous symmetric functions fh i : i 1g with nonnegative integer coefficients. An interesting combinatorial invariant of these homology representations is the multiplicity of the trivial representation (see [St1, Section 7] and [Su1, Sections 3 and 4]) If the preceding conjectures were true, the multiplicity 10 SHEILA SUNDARAM fi 2n;k of the trivial S 2n module in the homology of Pi e 2n ( k) would simply be the number of S 2n orbits. Note that from Theorem 2.1 we have fi 2n;n = n X i=2 b i (n) In particular the sum P n i=2 b i ....

....In particular, we have the formula Corollary 4.10. For symmetric functions f and g; h 2 [f g] h 2 [f ] h 2 [g] f Delta g: The interested reader can now recover (2.2) from the computation of (4.2) We record one more example of a plethystic computation which is frequently useful. See [Su1] for applications to computing the multiplicity of the trivial representation in homology modules. These results are easily derived from the properties developed in this section. Proposition 4.11. Let be an integer partition of n: Then (i) s [1 q] q r 1 Gammaq n Gammar 1 1 Gammaq ; ....

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2) (1994), 225-296.

....i1 ( Gamma1) i Gamma1 i 3 5 = h 1 : Let A n;k denote the degree n term in X i1 h i 2 4 k X i1 ( Gamma1) i Gamma1 i 3 5 : The identity (5.1) may thus be rewritten A n;1 = h 1 : We also deduce the formulas (5. 2) A n;k = 0; k n 2; A k 1;k = Gamma( Gamma1) k k 1 Recall from [S1] that if = Q i i m i n denotes an integer partition of n with m i parts of size i; then the degree n term in the left hand side of (5.1) is obtained by computing the following sum of products of plethysms: 5.3) X n = Q i i m i Y i hm i [ Gamma1) i Gamma1 i ] The key ....

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. in Math. 104 (2)(1994), 225-296.

No context found.

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math. 104 (1994), no. 2, 225-296.

No context found.

S. Sundaram, The homology representations of the symmetric group on Cohen-Macaulay subposets of the partition lattice, Adv. Math. 104 (1994), no. 2, 225-296.

No context found.

S. Sundaram, The homology representations of the symmetric group on CohenMacaulay subposets of the partition lattice, Adv. in Math. 104 (1994), no. 2, 225--296.

No context found.

S. Sundaram, The homology representations of the symmetric group on CohenMacaulay subposets of the partition lattice, Adv. in Math. 104 (1994), no. 2, 225--296.

No context found.

S. Sundaram, The homology representations of the symmetric group on CohenMacaulay subposets of the partition lattice, Adv. in Math. 104 (1994), no. 2, 225--296.

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