Reduced Normal Cartan Matrix

• Reduced Normal Cartan Matrix is the Smith normal form of the classical Cartan matrix, encoding the invariant factors and structure of the Cartan group.
• It quantifies the failure of the Cartan map to be surjective, providing a precise measure of how far an algebra is from being quasi-hereditary.
• In standardly stratified algebras, the Δ-Cartan matrix facilitates an explicit computation of torsion and free components, preserving derived invariance.

The reduced normal Cartan matrix of a finite-dimensional $k$-algebra $\Lambda$ is the Smith normal form over $\mathbb Z$ of its classical Cartan matrix $C_\Lambda$. It is a canonical diagonal form that encodes the invariant factors of the cokernel of the Cartan map, hence the structure of the Cartan group $G_\Lambda := \operatorname{coker}(C_\Lambda)$. In the setting of standardly stratified algebras, this invariant becomes especially explicit: it is controlled by the $\Delta$-Cartan matrix, is always finite, is preserved under derived equivalence, and provides a precise measure of how far $\Lambda$ is from being quasi-hereditary (Marcos et al., 2018).

1. Cartan matrix and integral structure

Let $\Lambda$ be a finite-dimensional $k$-algebra, and fix a complete set of pairwise non-isomorphic simple left $\Lambda$-modules $\Lambda$0, with corresponding indecomposable projective covers $\Lambda$1. The Grothendieck groups $\Lambda$2 and $\Lambda$3 are free abelian of rank $\Lambda$4, canonically identified with $\Lambda$5 via the bases $\Lambda$6 and $\Lambda$7, respectively.

The Cartan map is the $\Lambda$8-linear transformation

$\Lambda$9

where

$\mathbb Z$0

Its representing matrix in the chosen bases is the Cartan matrix

$\mathbb Z$1

with entries given by the multiplicity of $\mathbb Z$2 in a composition series of $\mathbb Z$3.

When $\mathbb Z$4 is elementary, meaning $\mathbb Z$5, one may choose a complete set of primitive idempotents $\mathbb Z$6 with $\mathbb Z$7. In that case,

$\mathbb Z$8

Thus the classical Cartan matrix is an explicit integer matrix determined by the dimensions of the corner spaces $\mathbb Z$9.

This integral description is fundamental because the reduced normal Cartan matrix is not defined over $C_\Lambda$0 or over $C_\Lambda$1, but over $C_\Lambda$2. Its role is to classify the failure of the Cartan map to be surjective, or bijective, at the level of Grothendieck groups.

2. Smith normal form and the Cartan group

For any integer matrix $C_\Lambda$3, there exist unimodular matrices $C_\Lambda$4 such that

$C_\Lambda$5

where $C_\Lambda$6 and the invariant factors $C_\Lambda$7 satisfy

$C_\Lambda$8

This is the Smith normal form. The resulting diagonal matrix is uniquely determined by $C_\Lambda$9 up to the invariant factors.

The reduced normal Cartan matrix of $G_\Lambda := \operatorname{coker}(C_\Lambda)$0 is defined to be the Smith normal form of $G_\Lambda := \operatorname{coker}(C_\Lambda)$1 over $G_\Lambda := \operatorname{coker}(C_\Lambda)$2:

$G_\Lambda := \operatorname{coker}(C_\Lambda)$3

Its invariant factors encode all torsion information in the associated cokernel (Marcos et al., 2018).

The associated Cartan group is

$G_\Lambda := \operatorname{coker}(C_\Lambda)$4

equivalently,

$G_\Lambda := \operatorname{coker}(C_\Lambda)$5

Using the Smith normal form, one obtains the standard decomposition

$G_\Lambda := \operatorname{coker}(C_\Lambda)$6

Hence $G_\Lambda := \operatorname{coker}(C_\Lambda)$7 is finite if and only if $G_\Lambda := \operatorname{coker}(C_\Lambda)$8, equivalently $G_\Lambda := \operatorname{coker}(C_\Lambda)$9. In that case,

$\Delta$0

For triangular matrices $\Delta$1 with $\Delta$2, if $\Delta$3, then the exponent $\Delta$4 is a multiple of each diagonal entry $\Delta$5. This fact becomes relevant for $\Delta$6-Cartan matrices, which are triangular in the standardly stratified setting.

The reduced normal Cartan matrix is therefore a complete integral invariant of the finitely generated abelian group $\Delta$7. It does not merely detect finiteness; it gives the exact torsion profile and any free $\Delta$8-summands.

3. Standardly stratified algebras and the $\Delta$9-Cartan matrix

Fix a linear order $\Lambda$0 on $\Lambda$1. The standard $\Lambda$2-modules are

$\Lambda$3

the largest quotient of $\Lambda$4 whose composition factors are $\Lambda$5 with $\Lambda$6. Writing $\Lambda$7, let $\Lambda$8 denote the full subcategory of $\Lambda$9-modules admitting a finite filtration with subquotients in $\Lambda$0. The algebra $\Lambda$1 is standardly stratified if $\Lambda$2.

In this setting one considers the $\Lambda$3-Cartan matrix

$\Lambda$4

A key structural fact is that $\Lambda$5 is upper triangular with diagonal entries

$\Lambda$6

Consequently,

$\Lambda$7

(Marcos et al., 2018).

The main theorem in this setting identifies the Cartan group with the cokernel of the $\Lambda$8-Cartan matrix:

$\Lambda$9

Moreover,

$k$0

and $k$1 is a multiple of each $k$2.

If $k$3 is elementary, then

$k$4

In particular, $k$5 is upper triangular with positive diagonal entries, so $k$6 is finite.

This identifies the reduced normal Cartan matrix as a stratification-sensitive refinement of the classical Cartan matrix. In standardly stratified situations, it is frequently easier to compute the Smith normal form through $k$7 than through $k$8 itself.

4. Weakly triangular and quasi-hereditary cases

A particularly transparent case occurs when

$k$9

Equivalently, $\Lambda$0 is weakly triangular with respect to $\Lambda$1. Then the $\Lambda$2-Cartan matrix is diagonal:

$\Lambda$3

Consequently,

$\Lambda$4

In this regime the invariant factors of the reduced normal Cartan matrix are read off directly from endomorphism dimensions of the standard modules (Marcos et al., 2018).

The quasi-hereditary criterion is equally sharp. One has

$\Lambda$5

Equivalently,

$\Lambda$6

or, in terms of the reduced normal Cartan matrix,

$\Lambda$7

so all invariant factors are equal to $\Lambda$8.

This provides an exact interpretation of the Cartan group in the standardly stratified setting: the size and torsion of $\Lambda$9, as read off from the reduced normal Cartan matrix, measure how far $\Lambda$00 is from being quasi-hereditary. The statement is structural rather than heuristic, because it is expressed by an if and only if criterion.

The mechanism behind finiteness is given by a factorization associated with an Ext-projective stratifying system $\Lambda$01 of size $\Lambda$02. If $\Lambda$03, then

$\Lambda$04

with $\Lambda$05 unimodular and triangular with determinant $\Lambda$06, while $\Lambda$07 is upper triangular with diagonal entries $\Lambda$08. Hence

$\Lambda$09

and $\Lambda$10, ensuring finiteness of the Cartan group.

5. Derived invariance and realization results

If $\Lambda$11 and $\Lambda$12 are derived equivalent algebras, then there exists $\Lambda$13 such that

$\Lambda$14

Since Smith normal form is invariant under left and right multiplication by unimodular matrices, the invariant factors $\Lambda$15 are preserved under derived equivalence. Therefore the reduced normal Cartan matrix and the Cartan group are derived invariants; in particular,

$\Lambda$16

(Marcos et al., 2018).

This invariance places the reduced normal Cartan matrix among the integral invariants of derived categories. It detects information that is invisible over $\Lambda$17, because the Smith normal form records torsion in the cokernel of the Cartan map.

The same framework also yields realization results. Given integers $\Lambda$18, there exists a standardly stratified algebra $\Lambda$19 with

$\Lambda$20

so that

$\Lambda$21

The construction uses a quiver with one vertex $\Lambda$22 carrying a loop $\Lambda$23 such that $\Lambda$24 has $\Lambda$25-dimension $\Lambda$26, realized via relations on powers of the loop, together with a stratifying linear order.

As a consequence, any finite abelian group can be realized as the Cartan group of some standardly stratified algebra. In terms of the reduced normal Cartan matrix, this means that arbitrary prescribed invariant factors can occur.

A second explicit family arises for radical square zero algebras. If $\Lambda$27 with $\Lambda$28 having no proper oriented cycles and all loops appearing only at quasi-sources, and if $\Lambda$29 is a linear refinement of the reachability preorder $\Lambda$30, then

$\Lambda$31

and hence

$\Lambda$32

This provides a broad class of standardly stratified algebras with completely explicit reduced normal Cartan matrices and Cartan groups.

6. Computation, examples, and related context

To compute

$\Lambda$33

one applies integer row and column operations corresponding to left and right multiplication by unimodular matrices. The permitted operations are: swapping two rows or columns, adding an integer multiple of one row to another or one column to another, and multiplying a row or column by $\Lambda$34. These operations preserve cokernels and Smith normal form equivalence (Marcos et al., 2018).

A standard procedure is as follows. First, choose a nonzero entry of minimal absolute value and bring it to position $\Lambda$35 by row and column swaps. Second, use column operations to reduce all entries in row $\Lambda$36 modulo $\Lambda$37, making them $\Lambda$38, and row operations to reduce all entries in column $\Lambda$39 modulo $\Lambda$40, making them $\Lambda$41. Third, if any entry in the first row or column is not divisible by $\Lambda$42, replace $\Lambda$43 with $\Lambda$44 using the Euclidean algorithm, and repeat the reduction step. Eventually the entire first row and column outside $\Lambda$45 become $\Lambda$46. One then recurses on the $\Lambda$47 submatrix and finally ensures the divisibility chain

$\Lambda$48

The final diagonal determines the structure of $\Lambda$49: zero diagonal entries correspond to free $\Lambda$50-summands, while positive diagonal entries yield cyclic torsion summands $\Lambda$51.

Two worked examples illustrate the process. If a standardly stratified algebra has

$\Lambda$52

then the Smith normal form is already diagonal, so the reduced normal Cartan matrix is $\Lambda$53 and

$\Lambda$54

Its order is $\Lambda$55, and its exponent is $\Lambda$56. In terms of standard modules,

$\Lambda$57

For a non-diagonal example, consider an algebra with classical Cartan matrix

$\Lambda$58

Subtracting column $\Lambda$59 from column $\Lambda$60 yields $\Lambda$61. Hence

$\Lambda$62

and

$\Lambda$63

This matches the computation referred to as Example 3.6.

The reduced normal Cartan matrix also sits in a broader network of invariants. For algebras of finite global dimension, $\Lambda$64, and it is conjectured that $\Lambda$65. In that situation the reduced normal Cartan matrix is unimodular and $\Lambda$66, reflecting that the Cartan map is an isomorphism on $\Lambda$67. The Cartan matrix also underlies the Euler characteristic and Coxeter transformation in Auslander–Reiten theory, although the focus here is specifically on $\Lambda$68. This suggests that the reduced normal Cartan matrix isolates the integral obstruction carried by the Cartan map while remaining tightly connected to classical homological and representation-theoretic structures.