Keywords

Mathematics Subject Classification (2010)

1 Introduction

To \(A_0,\dots ,A_\ell \in \operatorname {\mathrm {M}}_{d}(\mathbb {C})\) one assigns the determinantal hypersurface

$$\displaystyle \begin{aligned} \{[\xi_0:\cdots:\xi_\ell]\in\mathbb{C}\mathbb{P}^\ell\colon \det (\xi_0 A_0+\cdots+\xi_\ell A_\ell)=0\}. \end{aligned} $$
(1.1)

This is a classical object in algebraic geometry [1, 6, 10, 11], where a key question asks which hypersurfaces admit determinantal representations. When A j are real symmetric matrices, determinantal hypersurfaces pertain to hyperbolic and stable polynomials [2, 15, 18, 23, 24]. The geometry of the hypersurface (1.1) is also explored in multivariate operator theory [3, 4, 26]. If A j are bounded operators on a Hilbert space and the determinant in (1.1) is replaced with the condition that ξ 0 A 0 + ⋯ + ξ ℓ A ℓ is not invertible, then (1.1) is known as the projective joint spectrum of A 0, …, A ℓ (cf. Taylor spectrum [22] for ensembles of commuting operators).

Through the work of Frobenius [13] and Dedekind [7] on group determinants (see also [9]), determinantal hypersurfaces also pertain to representation theory. Several fascinating developments in this direction [5, 14, 21] have been recently made. This note addresses certain limitations for extensions of these results.

Let G be a finitely generated group. If T = (g 1, …, g ℓ) is a finite sequence of generators for G and \(\rho :G\to \operatorname {\mathrm {GL}}_d(\mathbb {C})\) is a representation of G, then denote

$$\displaystyle \begin{aligned} \mathscr{Z}_1(T,\rho) = \left\{\xi \in\mathbb{C}^\ell\colon \det\left(I_d+\xi_1\rho(g_1)+\cdots+\xi_\ell\rho(g_\ell)\right)=0 \right\}. \end{aligned} $$
(1.2)

It is natural to ask what kind of information the affine hypersurface \(\mathscr {Z}_1(T,\rho )\) carries about ρ and G. For example, if G 1, G 2 are finite groups with left regular representations λ 1, λ 2, then \(\mathscr {Z}_1(G_1\setminus \{1\},\lambda _1)=\mathscr {Z}_1(G_2\setminus \{1\},\lambda _2)\) implies that G 1, G 2 are isomorphic [12]. However, one is typically interested in smaller generating sets or in finitely generated groups which are not necessarily finite. In [14], the authors computed the joint spectrum for the infinite dihedral group

$$\displaystyle \begin{aligned}D_\infty=\langle a,t\mid a^2=t^2=1\rangle \end{aligned}$$

with respect to the generating set (1, a, t), and analyzed its properties through the representation theory of D ∞. Determinantal hypersurfaces also have a strong connection with representation theory in the case of finite Coxeter groups [5]. A Coxeter group is a finitely generated group on generators g 1, …, g ℓ satisfying

$$\displaystyle \begin{aligned}(g_ig_j)^{m_{ij}}=1 \end{aligned}$$

where m ii = 1 and m ij ≥ 2 for i ≠ j. In [5] the authors first showed that if G is a finite Coxeter group, λ is its left regular representation, and T = (g 1, …, g ℓ) are the generators as above, then \(\mathscr {Z}_1(T,\lambda )\) determines G up to isomorphism. Furthermore, if G is not of exceptional type (in the Coxeter diagram sense) and ρ is an arbitrary finite-dimensional representation of G, then \(\mathscr {Z}_1(T,\rho )\) determines ρ.

These theorems were presented during the Multivariable Spectral Theory and Representation Theory workshop at the Banff International Research Station in April 2019. Several problems about extending these results beyond Coxeter groups were posed by the speakers; among them were the following.

Questions 1.1

Let G be a finite group, T a fixed generating set for G, and ρ 1, ρ 2 irreducible complex representations of G.

  1. (1)

    Is \(\mathscr {Z}_1(T,\rho _1)\) a reduced and irreducible hypersurface?

  2. (2)

    If \(\mathscr {Z}_1(T,\rho _1)=\mathscr {Z}_1(T,\rho _2)\), are ρ 1 and ρ 2 equivalent?

As usual, \(\rho _1:G\to \operatorname {\mathrm {GL}}_{d_1}(\mathbb {C})\) and \(\rho _2:G\to \operatorname {\mathrm {GL}}_{d_2}(\mathbb {C})\) are equivalent if d 1 = d 2 and ρ 2 = Pρ 2 P −1 for some \(P\in \operatorname {\mathrm {GL}}_{d_1}(\mathbb {C})\). A representation ρ 1 is irreducible if its image does not admit a nontrivial common invariant subspace; equivalently, it generates \( \operatorname {\mathrm {M}}_{d_1}(\mathbb {C})\) as a \(\mathbb {C}\)-algebra by Burnside’s theorem [17, Corollary 1.17]. The hypersurface \(\mathscr {Z}_1(T,\rho _1)\) is reduced and irreducible (in the scheme-theoretic sense) if its defining determinant in (1.2) is an irreducible polynomial. The main result of this note is the following.

Theorem 1.2

Questions 1.1 (1) and (2) have negative answers in general.

See Sects. 2.1 and 2.2 for concrete examples. On a more positive side, in Sect. 3 we show that representation theory aligns well with a quantization of the determinantal hypersurface, the free locus; see Theorem 3.1. Furthermore, Proposition 3.4 determines whether a free locus arises from a representation of a finite group, and Proposition 3.7 characterizes finite abelian groups from the perspective of determinantal hypersurfaces. We conclude this note with an open question.

2 Representations Versus Determinants

In this section we give negative answers to Questions 1.1. The representations were found with the help of the computer algebra system GAP and the online repository ATLAS of Finite Group Representations. Verifying equivalence and irreducibility of representations was sometimes done symbolically with the computing system Mathematica.

2.1 Irreducible Representation with Reducible Determinant

The alternating group G = A 6 admits a presentation

$$\displaystyle \begin{aligned}G=\left\langle g_1,g_2\mid g_1^2, g_2^4, (g_1g_2)^5, (g_1g_2^2)^5 \right\rangle.\end{aligned}$$

Let

$$\displaystyle \begin{aligned}A_1 = \left(\begin{array}{rrrrrrrrr} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \end{array}\right)\end{aligned} $$

and

$$\displaystyle \begin{aligned}A_2 = \left(\begin{array}{rrrrrrrrr} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right).\end{aligned} $$

Then ρ(g 1) = A 1 and ρ(g 2) = A 2 determines a faithful and irreducible representation \(\rho :G\to \operatorname {\mathrm {GL}}_9(\mathbb {C})\). Indeed, we can directly check that

$$\displaystyle \begin{aligned}A_1^2=A_2^4=(A_1A_2)^5=(A_1A_2^2)^5=I,\end{aligned} $$

so ρ is a representation of G, and is moreover faithful since it is nontrivial and G is simple. Furthermore, all the possible products of A 1 and A 2 with at most 8 factors span the whole \( \operatorname {\mathrm {M}}_{9}(\mathbb {C})\), so ρ is irreducible. However, we claim that the curve \(\mathscr {Z}_1((g_1,g_2),\rho )\) in \(\mathbb {C}^2\) is not irreducible. We can compute the determinant of I + x 1 ρ(g 1) + x 2 ρ(g 2),

$$\displaystyle \begin{aligned}\det\left(\begin{array}{rrrrrrrrr} 1+x_1 & x_2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & x_1 & x_2 & 0 & 0 & 0 & 0 & 0 \\ 0 & x_1 & 1 & 0 & x_2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x_1 & x_2 & 0 & 0 & 0 \\ 0 & 0 & 0 & x_1 & 1 & 0 & x_2 & 0 & 0 \\ x_2 & 0 & 0 & 0 & 0 & 1 & 0 & x_1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1+x_1 & 0 & x_2 \\ 0 & 0 & 0 & 0 & 0 & x_1 & 0 & 1+x_2 & 0 \\ -x_1 & -x_1 & x_2-x_1 & -x_1 & -x_1 & -x_1 & -x_1 & -x_1 & 1-x_1 \\ \end{array}\right) \end{aligned}$$

by cofactor expansion along the rows. The reader will have no difficulty verifying that \(\det (I+x_1\rho (g_1)+x_2\rho (g_2))\) equals

$$\displaystyle \begin{aligned} &1+x_1-4 x_1^2-4 x_1^3+6 x_1^4+6 x_1^5-4 x_1^6-4 x_1^7+x_1^8+x_1^9+x_2+2 x_1 x_2-2 x_1^2 x_2 \\ &-6 x_1^3 x_2+6 x_1^5 x_2+2 x_1^6 x_2-2 x_1^7 x_2-x_1^8 x_2+x_1^2 x_2^2+x_1^3 x_2^2-2 x_1^4 x_2^2-2 x_1^5 x_2^2 \\ &+x_1^6 x_2^2+x_1^7 x_2^2-x_1^2 x_2^3+2 x_1^4 x_2^3-x_1^6 x_2^3-2 x_2^4+x_1^2 x_2^4-x_1^3 x_2^4+x_1^4 x_2^4+x_1^5 x_2^4 \\ &-2 x_2^5-2 x_1 x_2^5-x_1^2 x_2^5+x_1^4 x_2^5-x_1^2 x_2^6-x_1^3 x_2^6+x_1^2 x_2^7+x_2^8-x_1 x_2^8+x_2^9 \end{aligned}$$

which is the product of the following two irreducible polynomials:

$$\displaystyle \begin{aligned} & 1 + 2 x_1 - 2 x_1^3 - x_1^4 + x_1 x_2 + 2 x_1^2 x_2 + x_1^3 x_2 - x_1 x_2^2 - x_1^2 x_2^2 + x_1 x_2^3 - x_2^4, \\ & 1 - x_1 - 2 x_1^2 + 2 x_1^3 + x_1^4 - x_1^5 + x_2 - x_1 x_2 - x_1^2 x_2 + x_1^3 x_2 - x_2^4 - x_2^5. \end{aligned} $$

Some of the subsequent examples are presented in a more terse way to maintain the focus on their intent.

Note that the above irreducible representation of A 6 has dimension 9, which is not the minimum among nontrivial complex representations of A 6; namely, A 6 admits a representation σ of minimal dimension 5, and \(\mathscr {Z}_1((g_1,g_2),\sigma )\) is irreducible. One might thus be tempted to suggest that for a group G generated by a finite set T and its (irreducible) representation σ of minimal dimension, \(\mathscr {Z}_1(T,\sigma )\) is irreducible. However, even this weaker conjecture fails. The counterexample is given by the Janko group J 2,

$$\displaystyle \begin{aligned}J_2=\left\langle g_1,g_2\mid g_1^2, g_2^3, (g_1g_2)^7, (g_1g_2g_1^{-1}g_2^{-1})^{12}, (g_1g_2(g_1g_2g_1g_2^{-1})^2)^6 \right\rangle. \end{aligned}$$

This sporadic simple group of order 604800 admits two non-isomorphic complex representations σ 1, σ 2 of minimal dimension 14, courtesy of ATLAS of Finite Group Representations. As in the previous example (albeit with slightly longer calculations), one can explicitly check that the curve \(\mathscr {Z}_1((g_1,g_2),\sigma _1)=\mathscr {Z}_1((g_1,g_2),\sigma _2)\) has two irreducible components.

2.2 Non-equivalent Representations with the Same Determinant

The classical group \(G= \operatorname {\mathrm {GL}}_2(\mathbb {Z}/3\mathbb {Z})\) admits the presentation

$$\displaystyle \begin{aligned}G=\left\langle g_1,g_2,g_3\mid g_1^2, (g_1g_2^{-1})^2, (g_1g_3^{-1})^2, g_2^2g_3g_2^{-1}g_3, g_2g_3^2g_2g_3^{-1} \right\rangle. \end{aligned}$$

Let A 1, A 2, A 3 be the matrices

$$\displaystyle \begin{aligned}\begin{pmatrix} -\frac{1}{\sqrt{2}}& -\frac 12 - \frac{i}{2}\\ -\frac 12 + \frac{i}{2} & \frac{1}{\sqrt{2}} \end{pmatrix},\quad \begin{pmatrix} \frac 12 + \frac{i}{2}& \frac{1}{\sqrt{2}} \\ - \frac{1}{\sqrt{2}} & \frac 12 - \frac{i}{2} \end{pmatrix},\quad \begin{pmatrix} \frac 12 - \frac{i}{2} & \frac{i}{\sqrt{2}} \\ \frac{i}{\sqrt{2}} & \frac 12 + \frac{i}{2} \end{pmatrix}. \end{aligned}$$

There are faithful irreducible unitary representations \(\rho _+,\rho _-:G\to \operatorname {\mathrm {GL}}_2(\mathbb {C})\) given by

$$\displaystyle \begin{aligned}\rho_{\pm}(g_1) =\pm A_1,\quad \rho_{\pm}(g_2) =A_2, \quad \rho_{\pm}(g_3) = A_3. \end{aligned}$$

It is easy to check that ρ + and ρ − are not equivalent. On the other hand,

$$\displaystyle \begin{aligned}\mathcal{Z}_1((g_1,g_2,g_3),\rho_\pm)=\{(\xi_1,\xi_2,\xi_3)\colon 1 - \xi_1^2 + \xi_2 + \xi_2^2 + \xi_3 + \xi_3^2=0\}. \end{aligned}$$

3 Free Locus Perspective

In this section we will see how representations of a finitely generated group are determined by a noncommutative relaxation of (1.2). To \(A\in \operatorname {\mathrm {M}}_{d}(\mathbb {C})^\ell \) we associate the monic matrix pencil L A = I + A 1 x 1 + ⋯ + A ℓ x ℓ of size d in freely noncommuting variables x = (x 1, …, x ℓ). Thus L is an affine matrix over the free algebra \(\mathbb {C}\! \mathop {<}\! x \! \mathop {>}\). At a matrix point \(X\in \operatorname {\mathrm {M}}_{n}(\mathbb {C})^\ell \) it evaluates as

$$\displaystyle \begin{aligned}L_A(X)=I_{dn}+A_1\otimes X_1+\cdots+A_\ell\otimes X_\ell \in\operatorname{\mathrm{M}}_{dn}(\mathbb{C}). \end{aligned}$$

The free locus [19] of L A is the disjoint union of determinantal hypersurfaces

$$\displaystyle \begin{aligned}\mathscr{Z}(L_A)=\bigsqcup_{n\in\mathbb{N}}\mathscr{Z}_n(L_A),\quad \mathscr{Z}_n(L_A) = \left\{X \in\operatorname{\mathrm{M}}_{n}(\mathbb{C})^\ell\colon \det L_A(X)=0 \right\}. \end{aligned}$$

Given a group G generated by T = (g 1, …, g n) and a complex representation \(\rho :G\to \operatorname {\mathrm {GL}}_d(\mathbb {C})\), we write

$$\displaystyle \begin{aligned} \mathscr{Z}(T,\rho)=\mathscr{Z}\left(L_{\rho(g_1),\dots,\rho(g_\ell)}\right). \end{aligned} $$
(3.1)

By the definition of the free locus we see that (3.1) is indeed a quantization of (1.2). The existing results on free loci [16, 19] readily apply to group representations.

Theorem 3.1

For i = 1, 2 let G i be a group generated by a finite sequence T i and let ρ i be a complex representation of G i . Assume |T 1| = |T 2|.

  1. (1)

    If ρ i is irreducible, then there exists \(n_0\in \mathbb {N}\) such that \(\mathscr {Z}_n(T_1,\rho _1)\) is a reduced and irreducible hypersurface for all n ≥ n 0.

  2. (2)

    If ρ 1 and ρ 2 are irreducible, then \(\mathscr {Z}(T_1,\rho _1)=\mathscr {Z}(T_2,\rho _2)\) if and only if \(G_1/\ker \rho _1\cong G_2/\ker \rho _2\) and ρ 1, ρ 2 are equivalent.

  3. (3)

    For i = 1, 2 assume that G i is finite and ρ i is a faithful representation. Then \(\mathscr {Z}(T_1,\rho _1)=\mathscr {Z}(T_2,\rho _2)\) if and only if G 1≅G 2 via an isomorphism mapping T 1 to T 2.

Proof

  1. (1)

    A consequence of [16, Theorem 3.4].

  2. (2)

    A consequence of [19, Theorem 3.11].

  3. (3)

    Let \(\mathcal {T}_i\) be the \(\mathbb {C}\)-algebra generated by T i. Since G i is finite, its group algebra \(\mathbb {C} G_i\) is semisimple by Maschke’s theorem [17, Theorem 1.9]. Since \(\mathcal {T}_i\) is a quotient of \(\mathbb {C} G_i\), it is also semisimple. Then \(\mathscr {Z}(T_1,\rho _1)=\mathscr {Z}(T_2,\rho _2)\) if and only if T 1↦T 2 induces an algebra isomorphism \(\mathcal {T}_1\to \mathcal {T}_2\) by [19, Corollary 3.8]. This isomorphism then restricts to the group isomorphism G 1 → G 2.

    □

Remark 3.2

There is a deterministic bound on n 0 in Theorem 3.1(1) that is exponential in |T 1| and the dimension of ρ 1 by [16, Remark 3.5] (the bound is likely not optimal). Similarly, to verify \(\mathscr {Z}(T_1,\rho _1)=\mathscr {Z}(T_2,\rho _2)\) of Theorem 3.1(2,3), it suffices to check \(\mathscr {Z}_n(T_1,\rho _1)=\mathscr {Z}_n(T_2,\rho _2)\) for a fixed large enough n, exponential in |T i| and the dimension of ρ i by [19, Remark 3.7].

Free loci are defined for monic pencils with arbitrary matrix coefficients; we now describe how the geometry of the free locus \(\mathscr {Z}(L_A)\) detects whether the coefficients A 1, …, A ℓ generate a finite group. See also [8] for an efficient algorithm that determines finiteness of a finitely generated linear group.

Definition 3.3

Let \(\ell ,n\in \mathbb {N}\). Let \(C\in \operatorname {\mathrm {GL}}_n(\mathbb {Z})\) be the permutation matrix corresponding to the cycle (1 2 ⋯ n). If {1, …, n} = S 1 ⊔⋯ ⊔ S ℓ and P j is the orthogonal projection onto \( \operatorname {\mathrm {span}}\{e_k\colon k\in S_j\}\), then the matrix point

$$\displaystyle \begin{aligned}X=(P_1C,\cdots,P_\ell C)\in\operatorname{\mathrm{M}}_{n}(\mathbb{Z})^\ell \end{aligned}$$

is called a cycle partition. For given ℓ, n we thus have ℓ n cycle partitions.

Let \(\mu _\infty \subset \mathbb {C}\setminus \{0\}\) be the group of all roots of unity. The next proposition shows that if A 1, …, A ℓ generate a finite group, then \(\mathscr {Z}(L_A)\) intersects complex lines through cycle partitions only in points from μ ∞.

Proposition 3.4

Let \(A\in \operatorname {\mathrm {M}}_{d}(\mathbb {C})^\ell \) . Then A 1, …, A ℓ generate a finite group if and only if the following hold:

  1. (i)

    there is a positive definite \(P\in \operatorname {\mathrm {M}}_{d}(\mathbb {C})\) such that \(A_j^*PA_j=P\) for all j;

  2. (ii)

    for every cycle partition X and \(t\in \mathbb {C}\),

    $$\displaystyle \begin{aligned}tX\in\mathscr{Z}(L_A) \implies t\in \mu_\infty. \end{aligned}$$

Proof

(⇐) Every A j is invertible by (i). Let G be a group generated by A 1, …, A ℓ. Also by (i), G is a subgroup of the unitary group in \( \operatorname {\mathrm {GL}}_d(\mathbb {C})\) with respect to the inner product 〈u, v〉 = u ∗ Pv. Hence every element of G is diagonalizable. By [25, Corollary 4.9], a finitely generated subgroup of \( \operatorname {\mathrm {GL}}_d(\mathbb {C})\) is finite if and only if it is periodic (or torsion; i.e., every element has finite order). Since a diagonalizable matrix has a finite order if and only if all its eigenvalues lie in μ ∞, it suffices to verify that eigenvalues of every element of G lie in μ ∞.

To (i 1, …, i n) ∈{1, …, ℓ}n we associate the cycle partition \(X\in \operatorname {\mathrm {M}}_{n}(\mathbb {Z})^\ell \) by choosing S j = {e k: i k = j}. We claim that \(tX\in \mathscr {Z}(L_A)\) if and only if (−t)n is an eigenvalue of \(A_{i_1}\cdots A_{i_n}\). Indeed, using Schur complements it is easy to check that

$$\displaystyle \begin{aligned} \det(I-(-1)^nt^nA_{i_1}\cdots A_{i_n}) &=\det\begin{pmatrix} I & tA_{i_1} \\ (-1)^nt^{n-1}A_{i_2}\cdots A_{i_n} & I \end{pmatrix} \\ &=\det\begin{pmatrix} I & tA_{i_1} &0 \\ 0& I & tA_{i_2} & \\ -(-1)^nt^{n-2}A_{i_3}\cdots A_{i_n} & 0 & I \end{pmatrix} \\ & = \cdots \\ &=\det\begin{pmatrix} I & tA_{i_1} & & \\ & \ddots & \ddots & \\ & & I & tA_{i_{n-1}} \\ tA_{i_n} & & & I \end{pmatrix} \\ &=\det L_A(tX). \end{aligned} $$

Thus the matrix \(A_{i_1}\cdots A_{i_n}\) has finite order if and only if \(tX\in \mathscr {Z}(L_A)\) implies t ∈ μ ∞, which holds by (ii).

(⇒) If A 1, …, A ℓ generate a finite group G, then \(\mathbb {C}^d\) admits a G-invariant inner product

$$\displaystyle \begin{aligned}\langle u,v\rangle = \sum_{g\in G} (gu)^*(gv). \end{aligned}$$

If P is the positive definite matrix satisfying 〈u, v〉 = u ∗ Pv, then (i) holds. Furthermore, the proof of (ii) is already given in the previous paragraph. □

Remark 3.5

If additional information about A 1, …, A ℓ is given, say that their entries generate a number field (finite extension of \(\mathbb {Q}\)), then the size of the cycle partitions, which have to be tested in Proposition 3.4, can be bounded using Schur’s theorem on orders of finite matrix groups [17, Theorem 14.19].

Remark 3.6

Let \(p\in \mathbb {N}\) be prime. If μ ∞ in Proposition 3.4 is replaced by the group of power-of-p roots of unity, one obtains a free locus characterization of matrix tuples that generate a finite p-group.

We also show how the free locus certifies whether its defining coefficients generate a finite abelian group. The degree of an affine variety of codimension m is the number of intersection points of the variety with m hyperplanes in general position; in the case of a hypersurface, it is simply the degree of its square-free defining polynomial.

Proposition 3.7

Let G be a finite group generated by \(A_1,\dots ,A_\ell \in \operatorname {\mathrm {M}}_{d}(\mathbb {C})\) . Then G is abelian if and only if the irreducible components of \(\mathscr {Z}_n(L_A)\) have degree n for all \(n\in \mathbb {N}\).

Proof

Let \(\mathcal {A}\) be the \(\mathbb {C}\)-algebra generated by A 1, …, A ℓ. As in the proof of Theorem 3.1(3) we see that \(\mathcal {A}\) is semisimple. After a basis change (which does not affect the structure of G or \(\mathscr {Z}(L_A)\)) we can thus assume that

$$\displaystyle \begin{aligned}A_j=A_j^{(1)}\oplus\cdots\oplus A_j^{(s)} \end{aligned}$$

where \(A_1^{(k)},\dots ,A_\ell ^{(k)}\in \operatorname {\mathrm {M}}_{d_k}(\mathbb {C})\) determine an irreducible representation of G for every k = 1, …, s. For \(X\in \operatorname {\mathrm {M}}_{n}(\mathbb {C})^d\) let us view \(\det L_{A^{(k)}}(X)\) as a polynomial in the entries of X. If d k = 1, then \(\det L_{A^{(k)}}(X)\) is up to an affine change of coordinates equal to the determinant of a generic n × n matrix, and hence an irreducible polynomial of degree n. On the other hand, if d k > 1, then \(\det L_{A^{(k)}}(X)\) is a polynomial of degree d k n > n for all n, and irreducible for all large enough n by [16, Theorem 3.4]. Since G is abelian if and only if d 1 = ⋯ = d s = 1, and

$$\displaystyle \begin{aligned}\mathscr{Z}_n(L_A)=\mathscr{Z}_n(L_A^{(1)})\cup\cdots\cup \mathscr{Z}_n(L_A^{(s)}), \end{aligned}$$

it follows that G is abelian if and only if the irreducible components of \(\mathscr {Z}_n(L_A)\) are hypersurfaces of degree n. □

Remark 3.8

If ℓ = 2 and A 1, A 2 are hermitian, then \(\mathscr {Z}_1(L_A)\) alone determines whether G is abelian, cf. [20].

The last two propositions offer some directions for future research. Theorem 3.1 implies that the linear group G generated by a tuple A is determined by \(\mathscr {Z}(L_A)\). It would be interesting to know which properties of G can be deduced from the geometry of \(\mathscr {Z}(L_A)\). For example, intersections of \(\mathscr {Z}(L_A)\) with certain lines and hyperplanes determine whether G is finite or abelian. An open problem is how to decide whether a finite group G is nilpotent/solvable/simple (or any other group-theoretic property) by considering the geometry of the hypersurfaces \(\mathscr {Z}_n(L_A)\).