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, N^{2}, N^{3},
N^{4}, …

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 E_{ij}
for j
≤ h(i).)

A flag (F_{1}, F_{2}, …)
in n-dimensional space is a collection of n subspaces of C^{n} such
that each F_{i} is i-dimensional and each F_{i}
⊆ F_{i+1}.

**The
Hessenberg variety H(N,H)** is the collection of flags (F_{1}, F_{2},
…) in n-dimensional space such that

N F_{i}
⊆ F_{h(i)}

Nilpotent Hessenberg |
3 | 21 | 111 |

123 | 1+2q+2q^{2}+q^{3} |
1+2q | 1 |

The
Hessenberg varieties H(N,H) on this row are the collection of flags (F_{1},
F_{2}, F_{3}) such that N F_{i}
⊆ F_{i} 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 N^{2} is two-dimensional, and
the kernel of N^{3} 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 GL_{3}(C). For all of the subsequent tables, the
first row gives the Poincare polynomials of the associated Springer fibers.