Centralizer Matrix Algebras

• Centralizer matrix algebras are commutant algebras defined by the condition XA=AX, characterized by canonical block decompositions and explicit basis constructions.
• They encompass frameworks from Morita contexts to representation theory, leading to equivalence criteria and homological invariants such as derived and stable dimensions.
• Their study reveals strong rigidity and cellularity, underpinning applications in tensor models, symmetric groups, and quantum algebra via canonical decompositions.

Centralizer matrix algebras are commutant algebras attached to a fixed matrix, a family of matrices, or a representation inside a full matrix algebra. In the most classical form, for a field $k$ and $A\in M_n(k)$, the centralizer matrix algebra is

$C_k(A)=\{X\in M_n(k)\mid XA=AX\},$

also denoted $S_n(c,R)$ in related notation, and it admits the intrinsic description $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$ or, more generally, $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$ (Li et al., 24 Nov 2025, Wang, 2019). In other strands of the literature, closely related commutant phenomena are studied on generalized matrix algebras built from Morita contexts, on permutation centralizer algebras in symmetric-group settings, and on tensor-power centralizers such as $\operatorname{End}_G(V^{\otimes k})$ or $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$ (Fosner et al., 2014, Geloun et al., 2017, Barnes et al., 2013, Martirosyan et al., 2016). Across these settings, the central theme is rigidity: commutation constraints force explicit block structures, canonical bases, and strong homological restrictions.

1. Definitions and terminological scope

In the standard matrix-theoretic usage, a principal centralizer matrix algebra is

$S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$

for a matrix $c\in M_n(R)$ over a field or ring $A\in M_n(k)$0 (Xi et al., 2020, Li et al., 30 Sep 2025). This algebra is similarity-invariant: if $A\in M_n(k)$1, then $A\in M_n(k)$2 is isomorphic to $A\in M_n(k)$3 via $A\in M_n(k)$4, so its structure is controlled by the rational canonical form or Jordan form of the defining matrix (Li et al., 30 Sep 2025). A basic criterion isolates the cyclic case: $A\in M_n(k)$5 if and only if the characteristic polynomial equals the minimal polynomial, $A\in M_n(k)$6; in that case the centralizer is commutative (Li et al., 30 Sep 2025).

A second major usage arises from Morita contexts. If $A\in M_n(k)$7 is a Morita context, the associated generalized matrix algebra is

$A\in M_n(k)$8

with multiplication determined by the bimodule pairings (Fosner et al., 2014, Fadaee, 2021). In this setting, centralizer-type questions concern centralizing traces, Lie triple centralizers, and commutation modulo the center rather than only the commutant of a single matrix.

A third usage is representation-theoretic. For a finite group $A\in M_n(k)$9, the McKay centralizer algebra is

$C_k(A)=\{X\in M_n(k)\mid XA=AX\},$0

while in quantum-group settings one studies algebras such as

$C_k(A)=\{X\in M_n(k)\mid XA=AX\},$1

for $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$2, or $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$3 for $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$4 at roots of unity (Barnes et al., 2013, Martirosyan et al., 2016, Mazurenko et al., 2019). In symmetric-group and tensor-model contexts, permutation centralizer algebras are commutants of diagonal subgroup actions inside tensor powers of group algebras (Mattioli et al., 2016, Geloun et al., 2017).

2. Canonical forms, explicit bases, and dimension formulas

The fundamental structural description is module-theoretic. For $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$5, let $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$6 with $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$7-action $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$8. Then

$C_k(A)=\{X\in M_n(k)\mid XA=AX\},$9

and the primary decomposition of $S_n(c,R)$0 over $S_n(c,R)$1 yields a block decomposition of the centralizer (Wang, 2019). Over an algebraically closed field, if the Jordan blocks for eigenvalue $S_n(c,R)$2 have sizes

$S_n(c,R)$3

then the centralizer is block-diagonal by eigenvalues, and each $S_n(c,R)$4-block inside the $S_n(c,R)$5-primary component is upper-triangular Toeplitz of dimension $S_n(c,R)$6 (Wang, 2019). This gives Frobenius’s dimension formula

$S_n(c,R)$7

Over a general field, the rational canonical form replaces eigenvalues by irreducible polynomials $S_n(c,R)$8, and

$S_n(c,R)$9

where the $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$0 are the $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$1-primary sizes (Wang, 2019).

The same description yields explicit bases. For one Jordan block $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$2, the centralizer is $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$3 with basis $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$4. For multiple blocks, a concrete basis is given by block-supported matrices $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$5, where the $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$6-block is the truncation of $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$7 to the first $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$8 columns, with $C_k(A)\cong \operatorname{End}_{k[A]}(k^n)$9 (Wang, 2019). In rational canonical form, the analogous basis uses companion matrices $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$0, generators $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$1, and the block family $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$2 (Wang, 2019).

Generic-matrix methods isolate the extremal commutative case. If $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$3 has a cyclic vector, equivalently if the minimal polynomial has degree $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$4, then

$S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$5

If $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$6 is diagonalizable with distinct eigenvalues, then the centralizer consists of all diagonal matrices in the eigenbasis and is again equal to the polynomial algebra in $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$7 (Belov-Kanel et al., 2018). In the algebra of generic matrices $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$8, the minimal polynomial is irreducible, and for large prime size $S_n(c,R)\cong \operatorname{End}_{R[c]}(R^n)$9 it coincides with the characteristic polynomial; this forces generic centralizers to be polynomial algebras in the generic matrix (Belov-Kanel et al., 2018).

3. Matrix equivalences, singularity categories, and homological invariants

A central development is the reduction of categorical equivalences of centralizer matrix algebras to linear-algebraic invariants extracted from elementary divisors. Writing the minimal polynomial of $\operatorname{End}_G(V^{\otimes k})$0 as

$\operatorname{End}_G(V^{\otimes k})$1

one associates maximal divisors $\operatorname{End}_G(V^{\otimes k})$2, power-index sets $\operatorname{End}_G(V^{\otimes k})$3, and combinatorial transforms $\operatorname{End}_G(V^{\otimes k})$4 and $\operatorname{End}_G(V^{\otimes k})$5 of finite exponent sets $\operatorname{End}_G(V^{\otimes k})$6 (Li et al., 2023, Li et al., 30 Sep 2025). These data define matrix equivalence relations $\operatorname{End}_G(V^{\otimes k})$7, $\operatorname{End}_G(V^{\otimes k})$8, and $\operatorname{End}_G(V^{\otimes k})$9, and the principal classification theorem states: $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$0

$$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$1

$$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$2

(Li et al., 30 Sep 2025, Li et al., 2023). A later stable-equivalence classification introduces $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$3-equivalence on matrices and proves that, when the minimal polynomial of either matrix is separable, stable equivalence of centralizer matrix algebras is equivalent to $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$4-equivalence of the defining matrices (Li et al., 24 Nov 2025).

These equivalence criteria have strong homological consequences. Over perfect fields, stable equivalence for centralizer matrix algebras is equivalent to stable equivalence of Morita type, and then global dimension, finitistic dimension, and dominant dimension are preserved: $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$5

$$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$6

$$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$7

whenever $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$8 over a perfect field (Li et al., 24 Nov 2025). The same circle of ideas proves the Auslander–Reiten conjecture for algebras stably equivalent to a centralizer matrix algebra (Li et al., 24 Nov 2025).

More recent work describes singularity categories completely. For a centralizer algebra $$\operatorname{End}_{\mathbf U}(V_\lambda\otimes V^{\otimes n})$$9, the singularity category is a finite product of singularity categories of truncated polynomial algebras: $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$0 where the multiset $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$1 records the gaps in the exponent pattern of the primary decomposition (Chen et al., 21 Mar 2026). The corresponding $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$2-equivalence on matrices classifies singularity categories, while $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$3-equivalence classifies the algebras up to isomorphism (Chen et al., 21 Mar 2026). These results are then used to verify the Cartan determinant conjecture, the Auslander–Reiten/Gorenstein projective conjecture, and, consequently, all the classical homological conjectures listed in Auslander–Reiten–Smalø for centralizer matrix algebras over fields (Chen et al., 21 Mar 2026).

Permutation matrices form a particularly rigid subclass. Their elementary divisors are determined by cycle lengths and $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$4-adic valuations, and for them Morita equivalence and derived equivalence coincide (Li et al., 2023, Li et al., 30 Sep 2025). Derived equivalences of the centralizers of permutation matrices also descend to the $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$5-regular and $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$6-singular parts of the permutations (Li et al., 2023, Li et al., 30 Sep 2025).

4. Generalized matrix algebras, centralizing traces, and Lie-type maps

In generalized matrix algebras

$S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$7

centralizer phenomena are expressed through commutation modulo the center rather than only through equality $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$8 (Fosner et al., 2014). If $S_n(c,R):=\{X\in M_n(R)\mid Xc=cX\},$9 is $c\in M_n(R)$0-bilinear and $c\in M_n(R)$1 is its trace, the condition

$c\in M_n(R)$2

defines a centralizing trace (Fosner et al., 2014). Under $c\in M_n(R)$3-torsionfreeness, control of commuting linear maps on the diagonal corners, full visibility of the center through the corner projections, and loyalty of the off-diagonal bimodule $c\in M_n(R)$4, every such trace has the proper form

$c\in M_n(R)$5

where $c\in M_n(R)$6, $c\in M_n(R)$7 is linear, and $c\in M_n(R)$8 is a central-valued trace of a bilinear mapping (Fosner et al., 2014). In full matrix algebras this specializes to

$c\in M_n(R)$9

and in triangular algebras the same hypotheses force analogous quadratic-plus-linear-plus-central formulas (Fosner et al., 2014).

This rigidity feeds directly into Lie-theoretic classification. A Lie triple isomorphism $A\in M_n(k)$00 satisfies

$A\in M_n(k)$01

Under hypotheses mirroring those for centralizing traces, every Lie triple isomorphism is almost standard: $A\in M_n(k)$02 where $A\in M_n(k)$03 is a Jordan homomorphism and $A\in M_n(k)$04 is central-valued and vanishes on all second commutators (Fosner et al., 2014). In the case of $A\in M_n(k)$05, this recovers the familiar pattern $A\in M_n(k)$06, with $A\in M_n(k)$07 a Jordan homomorphism and $A\in M_n(k)$08 scalar-valued on the center (Fosner et al., 2014).

A parallel line studies Lie triple centralizers on generalized matrix algebras. A linear map $A\in M_n(k)$09 is a Lie triple centralizer if

$A\in M_n(k)$10

Such maps admit a complete blockwise description in terms of diagonal maps $A\in M_n(k)$11 and off-diagonal maps $A\in M_n(k)$12, with anchoring identities tying $A\in M_n(k)$13 and $A\in M_n(k)$14 to the diagonal components (Fadaee, 2021). Under annihilating hypotheses on the bimodules, proper Lie triple centralizers are exactly those of the form

$A\in M_n(k)$15

with $A\in M_n(k)$16 and $A\in M_n(k)$17 vanishing on all triple commutators (Fadaee, 2021). Closely related automatic-centralizer results also hold for weak $A\in M_n(k)$18-Jordan centralizers on generalized matrix algebras, triangular algebras, CSL algebras, and matrix algebras: under $A\in M_n(k)$19 and $A\in M_n(k)$20, every such map is a genuine centralizer $A\in M_n(k)$21 for some $A\in M_n(k)$22 (Guo et al., 2011).

5. Tensor-power centralizers and quantum-algebraic centralizers

In representation theory, centralizer algebras appear as commutants of tensor-power actions. For a finite subgroup $A\in M_n(k)$23 with defining representation $A\in M_n(k)$24,

$A\in M_n(k)$25

is finite-dimensional and semisimple, and Schur–Weyl theory gives

$A\in M_n(k)$26

(Barnes et al., 2013). The multiplicities $A\in M_n(k)$27 are the numbers of length-$A\in M_n(k)$28 walks from the affine node $A\in M_n(k)$29 to $A\in M_n(k)$30 in the McKay graph, so $A\in M_n(k)$31 is the number of length-$A\in M_n(k)$32 closed walks at $A\in M_n(k)$33 (Barnes et al., 2013). For $A\in M_n(k)$34, $A\in M_n(k)$35, and for finite $A\in M_n(k)$36 one has the inclusion $A\in M_n(k)$37 (Barnes et al., 2013).

Exceptional and quantum analogues preserve the commutant viewpoint. For $A\in M_n(k)$38, with $A\in M_n(k)$39 the $A\in M_n(k)$40-dimensional irreducible representation and $A\in M_n(k)$41 arbitrary irreducible, the centralizer algebra

$A\in M_n(k)$42

is generated by the image of the affine braid group $A\in M_n(k)$43 (Martirosyan et al., 2016). The generators are adjacent braidings $A\in M_n(k)$44 together with the affine element

$A\in M_n(k)$45

and the resulting surjection $A\in M_n(k)$46 realizes a Schur–Weyl-type duality for exceptional type $A\in M_n(k)$47 (Martirosyan et al., 2016).

At roots of unity, the quantum superalgebra $A\in M_n(k)$48 yields another family of centralizer algebras. For the typical module $A\in M_n(k)$49, the image of the braid generators $A\in M_n(k)$50 acting on $A\in M_n(k)$51 generates

$A\in M_n(k)$52

and the local generator satisfies the cubic polynomial

$A\in M_n(k)$53

(Mazurenko et al., 2019). The paper gives explicit bases for $A\in M_n(k)$54, $A\in M_n(k)$55, and $A\in M_n(k)$56, multiplication laws in the rank-three case, and the recursive span formula

$A\in M_n(k)$57

(Mazurenko et al., 2019).

6. Permutation centralizer algebras, matrix/tensor models, and monomial representations

Permutation centralizer algebras are commutants of subgroup actions inside symmetric-group algebras. In the two-matrix case, with $A\in M_n(k)$58,

$A\in M_n(k)$59

is a finite-dimensional semisimple algebra (Mattioli et al., 2016). Its Wedderburn–Artin decomposition is

$A\in M_n(k)$60

where $A\in M_n(k)$61 is the Littlewood–Richardson coefficient, so

$A\in M_n(k)$62

(Mattioli et al., 2016). Restricted Schur elements $A\in M_n(k)$63 furnish matrix units, and the center is generated by the block idempotents $A\in M_n(k)$64 (Mattioli et al., 2016).

For tensor models, the relevant algebra is the permutation centralizer algebra

$A\in M_n(k)$65

whose rank-$A\in M_n(k)$66 form $A\in M_n(k)$67 is semisimple and decomposes into full matrix blocks indexed by triples $A\in M_n(k)$68 with nonzero Kronecker coefficient $A\in M_n(k)$69 (Geloun et al., 2017). Its dimension is

$A\in M_n(k)$70

and the Fourier-transformed basis $A\in M_n(k)$71 satisfies the exact matrix multiplication rule

$A\in M_n(k)$72

(Geloun et al., 2017). These matrix units diagonalize Gaussian tensor-model two-point functions, and the center produces observables whose one-point functions are proportional to Kronecker coefficients (Geloun et al., 2017).

Witness-field versions of these constructions make the PCA structure constants directly visible in correlators. In matrix models, two-point functions with witness matrices $A\in M_n(k)$73 satisfy

$A\in M_n(k)$74

while in Schur or $A\in M_n(k)$75-bases one gets witness-generalized orthogonality relations of the form

$A\in M_n(k)$76

(Ramgoolam et al., 2023). The same pattern holds for two-matrix Necklace PCAs, multi-matrix PCAs, and tensor Kronecker PCAs, with the witness-only observable occupying the output slot of the orthogonality equation (Ramgoolam et al., 2023).

A different but related development studies centralizer algebras of monomial representations. For a monomial representation $A\in M_n(k)$77, the centralizer algebra $A\in M_n(k)$78 has a basis of orientable orbital matrices, where orientability is characterized by

$A\in M_n(k)$79

(Acevedo et al., 2024). In the multiplicity-free case, its character table is computed by double-coset character sums

$A\in M_n(k)$80

with $A\in M_n(k)$81 and

$A\in M_n(k)$82

(Acevedo et al., 2024). This framework contains group-developed and cocyclic-developed Hadamard matrices as centralizer-algebra constructions and supports Gröbner-basis classifications of highly symmetric complex Hadamard matrices (Acevedo et al., 2024).

7. Frobenius extensions, cellularity, and structural synthesis

Besides their module-theoretic and homological rigidity, centralizer matrix algebras often carry strong extension-theoretic structure. If $A\in M_n(k)$83 is Jordan-similar over an integral domain or algebraically closed field, then $A\in M_n(k)$84 is a cellular $A\in M_n(k)$85-algebra in the sense of Graham–Lehrer (Xi et al., 2020). In particular, if $A\in M_n(k)$86 is an algebraically closed field, every square matrix is Jordan-similar, so $A\in M_n(k)$87 is always cellular (Xi et al., 2020). Under additional hypotheses—either an invertible matrix with a $A\in M_n(k)$88-free point, or a Jordan-similar matrix over a ring without zero-divisors and with all eigenvalues in the center—one has that

$A\in M_n(k)$89

is a separable Frobenius extension (Xi et al., 2020).

A further refinement studies $A\in M_n(k)$90 as an extension of the base algebra $A\in M_n(k)$91. For a $A\in M_n(k)$92-algebra $A\in M_n(k)$93 and $A\in M_n(k)$94 with eigenvalues in $A\in M_n(k)$95, the restriction of the matrix trace

$A\in M_n(k)$96

defines a symmetric Frobenius functional, and $A\in M_n(k)$97 is a symmetric Frobenius extension (Wang et al., 19 Feb 2026). Over a field $A\in M_n(k)$98, separability of $A\in M_n(k)$99 is equivalent to diagonalizability of $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$00, equivalently to semisimplicity of the centralizer: $C_k(A)=\{X\in M_n(k)\mid XA=AX\},$01 (Wang et al., 19 Feb 2026). This criterion is the extension-theoretic counterpart of the classical distinction between semisimple block-diagonal centralizers and Toeplitz-type centralizers containing nilpotent radical.

Taken together, these results show that centralizer matrix algebras form a tightly organized class. In the single-matrix setting they are governed by rational and Jordan canonical forms, explicit Toeplitz or companion-matrix bases, and elementary-divisor combinatorics (Wang, 2019, Li et al., 30 Sep 2025). In generalized matrix algebras they control centralizing traces, Lie triple isomorphisms, and automatic-centralizer phenomena modulo the center (Fosner et al., 2014, Fadaee, 2021). In representation theory and mathematical physics they appear as semisimple commutants with explicit matrix-unit bases, whose block sizes are recorded by Littlewood–Richardson coefficients, Kronecker coefficients, path multiplicities on McKay graphs, or braid-group images (Barnes et al., 2013, Mattioli et al., 2016, Geloun et al., 2017, Martirosyan et al., 2016). A plausible implication is that “centralizer matrix algebra” is less a single construction than a recurring structural paradigm: once a commutation constraint is imposed inside a matrix or tensor category, the resulting algebra is often governed by canonical decomposition, double-centralizer behavior, and unexpectedly strong rigidity.