Julianna Tymoczko: algebraic and combinatorial geometer

Julianna Tymoczko

Associate Professor of Mathematics
Burton Hall 314
(413) 585–3775

Email: jtymoczko AT smith dot edu

Tables of Poincare Polynomials For Nilpotent Hessenberg Varieties (Type A)

This page gives tables of Poincare polynomials for the Hessenberg varieties H(N,H), which are read as follows.

N is a nilpotent n x n matrix.  Any such matrix can be expressed in Jordan canonical form, which partitions a complex n-dimensional vector space into blocks on which N is regular.  In these tables, N is denoted by its associated partition. 

Caution: The partition associated to N is labeled by the difference in size of the successive kernels N, N2, N3, N4, …

Thus, the nilpotent associated to 3 is the zero operator, while each regular nilpotent is associated to the partition 1 1 1.

H is a Hessenberg space, which is a vector space associated to a unique step function h: {1,2,…,n} → {1,2,…,n} satisfying the conditions that

h(i) ≥ max{i,h(i-1)} for each i.

In these tables, H is described as the sequence h(1) h(2) h(3) ….  For instance, the smallest possible Hessenberg space in three dimensions is 1 2 3, while the largest possible is 3 3 3.  (The terminology Hessenberg spaces arises because each of these step functions is associated to the linear subspace of n x n matrices generated by the matrix basis units Eij for j ≤ h(i).)

A flag (F1, F2, …) in n-dimensional space is a collection of n subspaces of Cn such that each Fi is i-dimensional and each Fi ⊆ Fi+1.

The Hessenberg variety H(N,H) is the collection of flags (F1, F2, …) in n-dimensional space such that

N Fi ⊆ Fh(i)

For example, the first row of the table for three-dimensional Hessenberg varieties reads



3 21 111


1+2q+2q2+q3 1+2q 1

The Hessenberg varieties H(N,H) on this row are the collection of flags (F1, F2, F3) such that N Fi ⊆ Fi for each i.  This condition is trivially satisfied when N is the zero matrix, which occurs in the first column.  When N is subregular – i.e., N has two Jordan blocks, one of which has dimension one – then the Hessenberg variety is homeomorphic to two spheres glued at a point.  The Poincare polynomial associated to that space is in the second column.  Finally, when N is regular, the kernel of N is one-dimensional, the kernel of N2 is two-dimensional, and the kernel of N3 is three-dimensional, so N is denoted by the partition 1 1 1.  There is exactly one flag in the Hessenberg variety H(N,H) in this case, so the Poincare polynomial in the third column is 1.

In fact, this row describes the Springer fibers in GL3(C).  For all of the subsequent tables, the first row gives the Poincare polynomials of the associated Springer fibers.

The Tables

  1. GL(2) (.7 KB)

  2. GL(3) (3.0 KB)

  3. GL(4) (19.8 KB)

  4. GL(5) (119.5 KB)

  5. GL(6) (808.1 KB)

  6. GL(7) (4.7 MB)

  7. GL(8) (29.4 MB)

  8. GL(9) (169.5 MB)