# 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

#### 12.0.1 two_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.