Wedderburn decomposition of the rational group algebras of $\operatorname{SL}_2(q)$ and $\operatorname{PSL}_2(q)$

1. Problem Statement

The paper addresses the classical problem in algebraic representation theory: determining the explicit Wedderburn decomposition of the rational group algebra $\mathbb{Q}G$ for finite groups of Lie type, specifically the special linear group $SL_2(q)$ and the projective special linear group $PSL_2(q)$, where $q = p^m$ is a prime power.

While the decomposition of group algebras over algebraically closed fields is well-understood via character theory, the decomposition over the rational field $\mathbb{Q}$ is significantly more complex. It requires determining:

1. The number of simple components.
2. The dimension of the matrix algebras ($n_i$).
3. The structure of the division rings ($D_i$) over their centers.
4. The Schur indices of the irreducible characters.

Existing methods often rely on computational tools (like GAP's WEDDERGA) or are limited to specific classes of groups (e.g., monomial groups). This paper aims to provide explicit combinatorial formulas depending solely on $q$ for these fundamental groups, covering all cases of $q$ (powers of 2 and powers of odd primes).

2. Methodology

The authors employ a combination of character theory, Galois theory, and Schur index theory to derive their results. The methodology proceeds as follows:

• Character Tables: The authors utilize the known complex character tables of $SL_2(q)$ and $PSL_2(q)$ (referencing standard texts like [11]). They analyze the values of irreducible characters $\chi$ to determine the field of character values $\mathbb{Q}(\chi)$.
• Galois Conjugacy: They group irreducible complex characters into Galois conjugacy classes over $\mathbb{Q}$. According to Schur theory, each distinct Galois conjugacy class corresponds to a unique irreducible rational representation.
• Schur Indices ($m_{\mathbb{Q}}(\chi)$): A critical step is determining the Schur index of each character over $\mathbb{Q}$.
  • For $PSL_2(q)$, they cite results showing $m_{\mathbb{Q}}(\chi) = 1$ for all $\chi$.
  • For $SL_2(q)$, they utilize a known table (Lemma 4.1) where Schur indices can be either 1 or 2, depending on the parity of $q$ and the specific character.
• Wedderburn Components: Using the formula for the dimension of the simple component associated with a character $\chi$:
  $$\text{dim}(A_{\mathbb{Q}(\chi)}) = m_{\mathbb{Q}}(\chi)^2 \cdot [\mathbb{Q}(\chi):\mathbb{Q}] \cdot \chi(1)^2$$
  The authors calculate the matrix size $n$ and the center of the division algebra $Z(D) = \mathbb{Q}(\chi)$.
• Division Algebra Identification: For cases where the Schur index is 2, the simple component is a matrix algebra over a non-commutative division ring (specifically a quaternion algebra). The authors explicitly identify these division algebras using cyclic algebra notation and Hilbert symbols, determining the specific parameters $(a, b)$ such that the algebra is isomorphic to $(a, b)_F$.

3. Key Contributions

The paper makes several distinct contributions to the literature:

1. Explicit Formulas: It provides closed-form combinatorial formulas for the Wedderburn decomposition of $\mathbb{Q}SL_2(q)$ and $\mathbb{Q}PSL_2(q)$ for all $q$. Previous works often required case-by-case computation or were limited to specific $q$.
2. Classification of Simple Modules: It determines the exact number of pairwise non-isomorphic simple $\mathbb{Q}G$-modules for every possible dimension.
3. Quaternion Algebra Identification: For $SL_2(q)$ with $q$ odd, the authors explicitly identify the quaternion algebras appearing in the decomposition, distinguishing between split algebras (isomorphic to matrix rings) and division algebras based on the Legendre symbol and the prime $p$.
4. Handling of $PSL_2(q)$ vs. $SL_2(q)$: The paper carefully distinguishes the decomposition of the projective group (where $Z(SL_2(q))$ is factored out) from the special linear group, particularly highlighting how the Schur indices change when moving from $PSL_2$ to $SL_2$.

4. Key Results

A. Case: $q$ is a power of 2

In this case, $SL_2(q) \cong PSL_2(q)$.

• Decomposition: The algebra decomposes into a direct sum of matrix algebras over cyclotomic fields $\mathbb{Q}(\zeta_d + \zeta_d^{-1})$ and the rational field $\mathbb{Q}$.
• Structure:
  $$\mathbb{Q}G \cong \mathbb{Q} \oplus M_q(\mathbb{Q}) \oplus \bigoplus_{d \in A} M_{q-1}(\mathbb{Q}(\zeta_d + \zeta_d^{-1})) \oplus \bigoplus_{d \in B} M_{q+1}(\mathbb{Q}(\zeta_d + \zeta_d^{-1}))$$
  where $A$ and $B$ are sets of divisors of $q+1$ and $q-1$ respectively. All Schur indices are 1.

B. Case: $q$ is a power of an odd prime $p$

Here, $SL_2(q) \neq PSL_2(q)$, and the results differ significantly.

1. For $PSL_2(q)$:

• All Schur indices are 1.
• The decomposition involves matrix algebras over $\mathbb{Q}$, $\mathbb{Q}(\sqrt{\pm q})$, and cyclotomic fields.
• Special Cases:
  • If $q \equiv 3 \pmod 4$: A component $M_{\frac{q-1}{2}}(\mathbb{Q}(\sqrt{-q}))$ appears.
  • If $q \equiv 1 \pmod 4$:
    • If $q$ is not a square: $M_{\frac{q+1}{2}}(\mathbb{Q}(\sqrt{q}))$ appears.
    • If $q$ is a square: Two copies of $M_{\frac{q+1}{2}}(\mathbb{Q})$ appear.

2. For $SL_2(q)$:

• Schur indices can be 1 or 2.
• The decomposition is more complex, involving quaternion division algebras when the Schur index is 2.
• Key Structural Differences from $PSL_2(q)$:
  • The sets of divisors $A, B$ are split into subsets $A, A'$ and $B, B'$ based on whether the divisor divides $q \pm 1$ or $(q \pm 1)/2$.
  • Quaternion Algebras:
    • For characters with odd indices, the division algebras are of the form $(\zeta_d - \zeta_d^{-1}, -q)_{\mathbb{Q}(\zeta_d + \zeta_d^{-1})}$ or similar.
    • For the "exceptional" characters (principal and complementary series), the division algebras depend on $q \pmod 4$ and whether $q$ is a square.
    • Example: If $q \equiv 1 \pmod 4$ and $q$ is a square, the component corresponding to the character of degree $(q-1)/2$ involves a quaternion algebra $D''$ defined over $\mathbb{Q}$ with parameters depending on $p$ (e.g., $(-1, -p)_{\mathbb{Q}}$ or $(-\ell, -p)_{\mathbb{Q}}$).

5. Significance

• Theoretical Completeness: The paper completes the classification of rational group algebras for the fundamental family of groups $SL_2(q)$ and $PSL_2(q)$, which are building blocks for all finite groups of Lie type.
• Computational Utility: By providing explicit formulas, the paper removes the need for heavy computational algebra systems for these specific groups, allowing for immediate theoretical analysis of their representation theory over $\mathbb{Q}$.
• Insight into Schur Indices: The work highlights the subtle behavior of Schur indices in $SL_2(q)$, demonstrating how the center of the group affects the rational representation theory, leading to the appearance of non-trivial division algebras (quaternion algebras) that do not appear in the projective case.
• Foundation for Further Research: The combinatorial techniques and explicit identification of division algebras serve as a template for studying rational group algebras of other finite groups of Lie type.

In summary, Choudhary and Panja have successfully translated the complex character data of $SL_2(q)$ and $PSL_2(q)$ into precise, combinatorial descriptions of their rational group algebras, resolving the structure of their simple components and the associated division rings for all prime powers $q$.