Matrices modulo $$p^k$$ and $$k$$ commuting matrices modulo $$p$$

1 Similarity and simultaneous similarity

1.1 Definition of Similarity

Matrices $$A$$ and $$B$$ in $$M_n(R)$$ are similar if there exists $$X\in GL_n(R)$$ such that $XAX^{-1} = B.$

Here $$R$$ can be any ring.

1.2 Simultaneous Similarity

The tuple $$(A_1, \dotsc, A_k)$$ is similar to the tuple $$(B_1,\dotsc, B_k)$$ if there exists $$X\in GL_n(R)$$ such that $XA_1X^{-1} = B_1, XA_2X^{-1} = B_2, \cdots, XA_kX^{-1} = B_k.$

2 Classes in a Field

2.1 Rational Canonical Form

$A \sim C_{f_1}\oplus C_{f_2} \oplus \dotsb \oplus C_{f_k}$

where $$f_k|f_{k-1}|\dotsc|f_1$$.

Here $$C_f$$ is the companion matrix of $$f$$.

2.2 In a finite field

Let $$\lambda = (\lambda_1,\dotsc, \lambda_k)$$ be a partition of $$n$$.

Number of classes with $$\deg(f_i) = \lambda_i$$ is $q^{\lambda_k + (\lambda_{k-1} - \lambda_k) + \dotsb (\lambda_2 - \lambda_1)} = q^{\lambda_1}$

Why?

$$f_1,\dotsc, f_k$$ is determined by $$f_1/f_2, f_2/f_3,\dotsc, f_{k-1}/f_k, f_k$$.

No. of choices for $$f_i/f_{i+1} = q^{\lambda_i - \lambda_{i+1}}$$.

3 Matrix tuple problem

$a_{n,k}(q) = \begin{array}{l}\text{No. of simultaneous similarity classes of}\\ \text{$$k$$-tuples of $$n\times n$$ matrices over \mathbf F_q}.\end{array}$

3.1 Burnside's Lemma:

$a_{n,k}(q) = \frac 1{|GL_n(\mathbf F_q)|}\sum_{g\in GL_n(\mathbf F_q)} |Z_{M_n(\mathbf F_q)}(g)|^k.$

Shows that

$A_n(q, t) = \sum_{k=0}^\infty a_{n,k}(q)t^k = \frac 1{|GL_n(\mathbf F_q)|}\sum_{g\in GL_n(\mathbf F_q)} \frac 1{1-|Z_{M_n(\mathbf F_q)}(g)|t},$

is a rational function of $$t$$ for each $$n$$.

4 Theory of Types

4.1 Jordan normal form

Similarity classes in $$M_n(\mathbf F_q)$$ correspond to $\{\phi : \mathrm{Irr}\mathbf F_q[t] \to \Lambda\mid \sum_{f\in\mathrm{Irr}\mathbf F_q[t]} \deg(f)|\phi(f)| = n\}$

4.2 Types

Classes $$\phi_1$$ and $$\phi_2$$ have the same type if

$\phi_2 = \phi_2\circ \sigma$

for some degree-preserving bijection $$\sigma: \mathrm{Irr}\mathbf F_q[t]\to \mathrm{Irr}\mathbf F_q[t]$$.

5 Why types are useful

5.1 Types are combinatorial

• Given $$\phi$$, define $$\tau:\Lambda\to \Lambda$$ by $\tau_\phi(\lambda) = (1^{m_1}, 2^{m_2}, \dotsc)$ where $$m_i$$ is the number of irreducible polynomials $$f$$ of degree $$i$$ such that $$\phi(i) = \lambda$$. Then $$\phi_1$$ and $$\phi_2$$ have the same type if and only if $$\tau_{\phi_1} = \tau_{\phi_2}$$.
• The set of types is a combinatorial object; types correspond to functions $\{\tau: \Lambda\to \Lambda \mid \sum_{\lambda\in \Lambda} |\lambda||\tau(\lambda)| = n\}$
• The number of matrices of each type can be counted
• Matrices of the same type have isomorphic centralizers

6 Computer Implementation

sage: from sage.combinat.similarity_class_type import *
sage: q = ZZ['q'].gen()
sage: def simultaneous_similarity_classes(n,k):
....:     return SimilarityClassTypes(n).sum(lambda la: q**(k*centralizer_algebra_dim(la)), invertible = True)/order_of_general_linear_group(n)
sage: simultaneous_similarity_classes(3, 2)
q^10 + q^8 + 2*q^7 + 2*q^6 + 2*q^5 + q^4

7 Kac conjecture

• Usually stated in the framework of quivers
• When the quiver has one vertex and $$k$$ loops (bouquet quiver) isomorphism classes of representations are simultaneous similarity classes of matrices

7.1 Conjecture (V. Kac, 1983)

$$a_{n,k}(q)$$ is a polynomial in $$q$$ with non-negative integer coefficients

This result was proved by Hausel, Letellier and Rodriguez-Villegas, 2013.

8 Commuting Matrix Tuple Problem

$b_{n,k}(q) = \begin{array}{l}\text{No. of simultaneous similarity classes of}\\ \text{$$k$$-tuples of commuting $$n\times n$$ matrices over \mathbf F_q}.\end{array}$

8.1 Generating function

$B_n(q, t) = \sum_{k = 0}^\infty b_{n,k}(q) t^k$

This is a rational function.

8.2 Not known

Are its coefficients polynomials in $$q$$?

9 Explicit Computation (Uday Bhaskar Sharma)

$$n$$ $$B_n(q,t)$$
1 $$\frac{1}{1-qt}$$

2 $$\frac{1}{(1-qt)(1-q^2t)}$$

3 $$\frac{1+q^2t^2}{(1-qt)(1-q^2t)(1-q^3t)}$$

4 $$\left(\frac{1+q^2t+2q^2t^2+q^3t^2+2q^4t^2+q^6t^3}{(1-qt)(1-q^2t)(1-q^3t)(1-q^4t)(1-q^5t)} \right)$$
$$- \left(\frac{q^5t+q^7t^2+q^3t^3+2q^7t^3+2q^9t^3+q^{10}t^4}{(1-qt)(1-q^2t)(1-q^3t)(1-q^4t)(1-q^5t)} \right)$$

9.1 Conclusion

$$b_{n,k}(q)$$ is a polynomial in $$q$$ with non-negative integer coefficients for $$n\leq 4$$.

10 Similarity classes modulo $$p^k$$

$c_{n,k}(q) = \text{No. of similarity classes of $$n\times n$$ matrices in M_n(\mathbf Z/p^k \mathbf Z)}.$

10.1 Generating Function

$C_n(q, t) = \sum_{k=0}^\infty c_{n,k}(q)t^k$

is known to be a rational function of $$t$$.

11 Calculations for $$k = 2$$ (Prasad, Singla and Spallone)

$$n$$ $$c_{n,2}(q)$$
$$2$$ $$q^4 + q^3 + q^2$$

$$3$$ $$q^6+q^5+2q^4+q^3+2q^2$$

$$4$$ $$q^8 + q^7 + 3q^6 + 3q^5 + 5q^4 + 3q^3 + 3q^2$$

12 Relation to simultaneous similarity classes

12.0.1two_theorem

$b_{n,2}(p) = c_{n,2}(p).$ See Singla, Jambor and Plesken and also: Prasad, Singla and Spallone, Remark 1.1

13 Outline of Proof

13.0.1 Main Lemma   Main_Lemma

For every $$A\in M_n(\mathbf Z/p\mathbf Z)$$, there exists $$\tilde A in M_n(\mathbf Z/p^2\mathbf Z)$$ such that for every $$B\in M_n(\mathbf Z/p\mathbf Z)$$ that commutes with $$A$$ there exists $$\tilde B in M_n(\mathbf Z/p^2\mathbf Z)$$ that commutes with $$\tilde A$$.

14 A reduction

Let $$A\in M_n(\mathbf Z/p\mathbf Z)$$.

Define $G_A = \{\tilde X \in GL_n(\mathbf Z/p^2\mathbf Z)\mid \tilde X\tilde A \cong \tilde A\tilde X \mod p\}.$

The map: $C\mapsto C\cap \tilde A + pM_n(\mathbf Z/p\mathbf Z)$

defines a bijection from the set of similarity classes in $$M_n(\mathbf Z/p^2\mathbf Z)$$ which which contain a lift of $$A$$ to the set of $$G_A$$-orbits in $$\tilde A + pM_n(\mathbf Z/p\mathbf Z)$$.

So, in order to classify similarity classes in $$M_n(\mathbf Z/p^2\mathbf Z)$$, it suffices to

• classify similarity classes in $$M_n(\mathbf Z/p\mathbf Z)$$ (which has been done)
• for some $$A$$ in each such class, find $$G_A$$-orbits in $$\tilde A + pM_n(\mathbf Z/p\mathbf Z)$$

15 A lemma on group actions

15.1 The lemma

Let $$G$$ be a group acting on a set $$X$$, and let $$N$$ be a normal subgroup of $$G$$.

Then $$G/N$$ has a well-defined action on $$N\backslash X$$, and

$G\backslash X = (G/N)\backslash (N\backslash X).$

15.2 In our case

$$G = G_A$$, $$X = \tilde A + pM_n(\mathbf Z/p\mathbf Z)$$.

$$N = \{I + pX\mid X\in M_n(\mathbf Z/p\mathbf Z)$$.

$$G/N$$ is isomorphic to $$Z_{GL_n(\mathbf Z/p\mathbf Z)}A$$.

16 $$N$$-orbits in $$X$$

$$\tilde A + p X$$ and $$\tilde A + pY$$ are in the same $$N$$ orbit if and only if there exists $$U\in M_n(\mathbf Z/p\mathbf Z)$$ such that

$(I + pU)(\tilde A + pX) = (\tilde A + pY)(I + pU).$

Equivalently

$X - Y \in [A, M_n(\mathbf Z/p\mathbf Z)].$

So: $N\backslash X = M_n(\mathbf Z/p\mathbf Z)/ [A, M_n(\mathbf Z/p\mathbf Z)]$

17 Duality

Identify $$M_n(\mathbf Z/p\mathbf Z)$$ with its linear dual using the non-degenerate bilinear form $\langle X, Y\rangle = \mathrm{trace}(XY).$

This form is invariant under the action of $$GL_n(\mathbf Z/p\mathbf Z)$$, and so also the action of $$G/N = Z_{GL_n(\mathbf Z/p\mathbf Z)}(A)$$.

This gives rise to an isomorphism

$\left( \frac{M_n(\mathbf Z/p\mathbf Z)}{[A, M_n(\mathbf Z/p\mathbf Z)]} \right)^* = Z_{M_n(\mathbf Z/p\mathbf Z)}(A)$

which preserves the action of $$Z_{GL_n(\mathbf Z/p\mathbf Z)}(A)$$.

18 Group actions on vector spaces and their duals

Let $$X$$ be a finite vector space, and $$G$$ be a finite group acting by linear maps.

Then ${}|G\backslash X | = | G\backslash X^*|$

This follows from Burnside's lemma:

${}|G\backslash X | = \frac 1{|G|} \sum_{g\in G} |X^g|$

Now note that $$X^g$$ is the $$1$$-eigenspace of $$g$$, which has the same dimension as the $$1$$-eigenspace of $$g^*$$.

19 Finishing the proof

So the number of similarity classes in $$M_n(\mathbf Z/p^2\mathbf Z)$$ which contain an element congruent to $$A\mod p$$ is

$Z_{GL_n(\mathbf Z/p\mathbf Z)}(A) \backslash \Big(M_n(\mathbf Z/p\mathbf Z)/[A, M_n(\mathbf Z/p\mathbf Z)]\Big)$

which is the same as $Z_{GL_n(\mathbf Z/p\mathbf Z)}(A) \backslash Z_{M_n(\mathbf Z/p\mathbf Z)}(A)$

So the number $$c_{n,k}(p)$$ of similarity classes in $$M_n(\mathbf Z/p^\mathbf Z)$$ is

$\sum_A |Z_{GL_n(\mathbf Z/p\mathbf Z)}(A) \backslash Z_{M_n(\mathbf Z/p\mathbf Z)}(A)|$

the sum being over all similarity classes in $$M_n(\mathbf Z/p\mathbf Z)$$.

This is $$b_{n,k}(p)$$.

20 Conjectures

• $$B_n(p, t) = C_n(p, t)$$ for all $$n$$.

This is known for $$n\leq 3$$, by work of Avni, Onn, Prasad and Vaserstein

• these are polynomials in $$p$$ with non-negative integer coefficients.

Date: 24 April 2015

Created: 2015-04-24 Fri 11:17

Emacs 24.3.1 (Org mode 8.2.4)

Validate