Central Cocharacter Sequences in PI-Algebras

• Central Cocharacter Sequences are algebraic constructs that record the decomposition of multilinear polynomial spaces into irreducible Sₙ representations, capturing central polynomial multiplicities.
• They provide a framework for classifying PI-algebras by linking exact sequences and bounded colengths to representation-theoretic invariants.
• By leveraging combinatorial tools and highest-weight theory, these sequences facilitate precise identification of irreducible components in the study of polynomial identities.

A central cocharacter sequence encodes, for an associative algebra with polynomial identities (PI-algebra) over a field of characteristic zero, the decomposition of the space of multilinear polynomials modulo central polynomials as an $S_n$-module, tracking the multiplicities of irreducible symmetric group representations. Central cocharacter sequences, along with their ordinary and proper central analogues, are fundamental invariants in the study of polynomial identities and their connections to the representation theory of symmetric and general linear groups.

1. Multilinear Polynomial Spaces and Central Polynomials

Let $F$ be a field of characteristic zero and $F\langle X\rangle$ denote the free associative $F$-algebra on a set of variables $X = \{x_1, x_2, \ldots \}$, with $$P_n = \mathrm{span}_F\{x_{\sigma(1)}x_{\sigma(2)}\ldots x_{\sigma(n)} : \sigma \in S_n\}$$ the space of all multilinear polynomials of degree $n$. For an associative $F$-algebra $A$:

• $\Id(A)$ is the T-ideal of identities: polynomials vanishing identically on $A$.
• $\Id^z(A)$ is the T-space of central polynomials: polynomials whose values always lie in $Z(A)$.

Key quotient spaces are defined as

• $P_n(A) = P_n/(P_n\cap\Id(A))$,
• $P_n^z(A) = P_n/(P_n\cap\Id^z(A))$,
• $\Delta_n(A) = (P_n\cap\Id^z(A))/(P_n\cap\Id(A))$.

There is an exact sequence of $S_n$-modules: $$0 \longrightarrow \Delta_n(A) \longrightarrow P_n^z(A) \longrightarrow P_n(A) \longrightarrow 0$$ implying $$\dim_F P_n(A) = \dim_F P_n^z(A) + \dim_F \Delta_n(A)$$.

2. Cocharacters, Central Cocharacters, and Multiplicities

The above spaces inherit an $S_n$-module structure by permutation of variables. Their characters are:

• $\chi_n(A)$: ordinary $n$th cocharacter of $A$ (for $P_n(A)$).
• $\chi_n^z(A)$: central cocharacter (for $P_n^z(A)$).
• $\chi_n^\delta(A)$: proper-central cocharacter (for $\Delta_n(A)$).

These admit multiplicity decompositions: $$\chi_n(A) = \sum_{\lambda \vdash n} m_\lambda \chi_\lambda, \quad \chi_n^z(A) = \sum_{\lambda \vdash n} m_\lambda^z \chi_\lambda, \quad \chi_n^\delta(A) = \sum_{\lambda \vdash n} m_\lambda^\delta \chi_\lambda$$ with $\chi_\lambda$ irreducible $S_n$-characters indexed by partitions $\lambda \vdash n$. Multiplicities satisfy $m_\lambda = m_\lambda^z + m_\lambda^\delta$; the colengths, central colengths, and proper central colengths are

$$l_n(A) = \sum_\lambda m_\lambda, \quad l_n^z(A) = \sum_\lambda m_\lambda^z, \quad l_n^\delta(A) = \sum_\lambda m_\lambda^\delta.$$

3. Explicit Central Cocharacter Sequences: Key Examples

Central cocharacter sequences can be calculated explicitly for major families of PI-algebras. The table below summarizes the structure for several archetypes:

Algebra	$\chi_n^z(A)$ Formula	$l_n^z(A)$ (for large $n$)
$UT(d_1,...,d_k),\;k>1$	$\chi_n^z = \chi_n$	$=l_n(A)$
Grassmann $\G_{2k}$	$\sum_{i=0}^{k-1} \chi_{(n-i,1^{2i})}$	$k$
$N_3$	$\chi_{(n)}$	1
$UT_2$	$\chi_n^z = \chi_n$	$=l_n(A)$
$A_4$, $A_4^*$ (one nonzero entry in block)	$\chi_n^z = \chi_n$	5
$A_5$ ($\dim Z=1$)	$\chi_{(n)} + 2\chi_{(n-1,1)}$	3
$A_2 \oplus A_1$	$\chi_{(n)} + \chi_{(n-1,1)}$	2
$\G_4 \oplus A_1$	$\chi_{(n)} + \chi_{(n-1,1)} + \chi_{(n-2,1^2)}$	3

• For $UT(d_1,...,d_k)$ with $k>1$, all central polynomials are identities so $P_n^z(A) = P_n(A)$, $\Delta_n(A)=0$.
• In finite Grassmann algebras, central and proper central cocharacters split into even- and odd-leg partitions, respectively.
• In the algebra $N_3$, only the trivial partition $(n)$ supports a central polynomial, while the proper central part is built from the next two partitions.

4. Classification of PI-Algebras by (Central) Colengths

A comprehensive classification, up to PI/T-equivalence, is achieved for algebras with bounded colength or central colength:

• If $l_n(A) \leq 6$ for all large $n$, then $A\sim_T N\oplus B$, with $B$ a basic algebra from the set

$\left\{ C\ (\text{commutative non-nilpotent}),A_1,A_1^*,A_1\oplus A_1^*,A_2, A_1\oplus A_2,A_1^*\oplus A_2, A_4,A_4^*,A_4\oplus A_1^*,A_4^*\oplus A_1,A_5,A_6,\G_4,\G_4\oplus A_1,\G_4\oplus A_1^* \right\}$

• For $l_n^z(A)\leq 2$ for all large $n$, $A\sim_T N \oplus B$ with $B$ drawn from

$\left\{ C, A_1, A_1^*, A_2, A_2\oplus A_1, A_2\oplus A_1^*, \G_4 \right\}$

and $N$ nilpotent. Such a classification excludes other PI-varieties from exhibiting bounded central colength.

5. Combinatorial and Representation-Theoretic Tools

The calculation of cocharacter and central cocharacter sequences leverages the interplay of symmetric group and general linear group representation theory. Key tools include:

• Identification of irreducible $S_n$-characters $\chi_\lambda$ with partitions $\lambda\vdash n$.
• Highest-weight theory for $GL_m$, identifying the occurrence and multiplicity of partitions via highest-weight vectors and multitableaux.
• The construction of highest-weight vectors $f_{T_\lambda}$ through standard polynomials in the columns of Young diagrams.
• The use of the Poincaré–Birkhoff–Witt basis and specific evaluations to test the non-vanishing of highest-weight vectors.

Multiplicity calculations reduce to checking whether a constructed highest-weight polynomial is nonzero in $A$; such arguments ascertain which $\lambda$ occur in the decomposition for a given algebra.

6. Structure of the Exact Sequences and their Implications

The relationship between $P_n(A)$, $P_n^z(A)$, and $\Delta_n(A)$ is governed by the exact sequence of $S_n$-modules. The additive identities at the level of characters and dimensions directly relate central, ordinary, and proper central invariants: $$\chi_n(A) = \chi_n^z(A) + \chi_n^\delta(A),\qquad m_\lambda = m_\lambda^z + m_\lambda^\delta$$ This structure allows for fine-grained tracking of the emergence and disappearance of central and proper central polynomials within multilinear identities, reflecting symmetry and centralizer properties in the PI-algebra.

7. Central Cocharacter Sequences in the Broader Context of PI-Theory

Central cocharacter sequences serve as invariants distinguishing PI-varieties, guiding classification theorems and supporting explicit identification of algebraic identities and central behaviors. Their role is both computational and conceptual, as they connect explicit polynomial constructions with deeper aspects of noncommutative invariant theory, the structure of central polynomials, and the nature of irreducible $S_n$-module decomposition in PI-algebra settings (Cota et al., 14 Jan 2026).