Pieri Rules for Classical Groups

• Pieri rules for classical groups are explicit combinatorial formulas that decompose product representations and cohomology classes for types A, B, C, and D.
• They use methods like horizontal and vertical strips along with specialized tableaux to translate tensor products and Schubert cycle multiplications into canonical basis expansions.
• Extensions in quantum cohomology and K-theory incorporate recursive combinatorial models and branching laws, providing actionable insights into representation theory and symmetric functions.

The Pieri rules for classical groups constitute a powerful family of combinatorial formulas describing the decomposition of product representations or cohomology classes associated with classical Lie groups of types A, B, C, and D. These rules encode how special representations—typically symmetric or exterior powers of the defining representation—tensor with arbitrary irreducibles, or how multiplication by distinguished Schubert cycles or symmetric functions expands into canonical bases. The rules manifest in representation theory, Schubert calculus, K- and quantum theory, and in the structure of symmetric functions and their generalizations, always with precise, explicit combinatorics governing the allowed summands.

1. Classical and Quantum Pieri Rules: Definitions and Formulations

For $GL_n(\mathbb{C})$ (Type A), the Pieri rule gives the irreducible decomposition of products such as $\Pi_\lambda\otimes\mathrm{Sym}^r(V)$ and $\Pi_\lambda\otimes\wedge^r(V)$. Explicitly, this corresponds to the expansion of a Schur function $s_\lambda$ times $h_r$ or $e_r$ into Schur functions $s_\nu$, with $\nu/\lambda$ a horizontal (resp. vertical) $r$-strip: $$s_\lambda\,h_r = \sum_{\nu:\,\nu/\lambda\;\text{hor.}\; r\text{-strip}} s_\nu\,, \qquad s_\lambda\,e_r = \sum_{\nu:\,\nu/\lambda\;\text{vert.}\; r\text{-strip}} s_\nu\,.$$ This classical paradigm extends to quantum and $K$-theoretic contexts:

• In the (small) quantum cohomology $QH^*(X)$ for Grassmannians and isotropic Grassmannians (types B, C, D), the quantum Pieri rule expresses the quantum product of a special Schubert class (often the Chern class $c_p(S^*)$ of the tautological bundle) with any Schubert class as a sum over classical-type terms (indexed by admissible diagram moves) and quantum corrections, determined by specific degenerations and Gromov-Witten invariants (Leung et al., 2013).
• In $K$-theory, the Pieri rules involve signed coefficients and tableau enumeration, for instance, using KOG-tableaux and KLG-tableaux for types D and C (Buch et al., 2010, Arroyo, 7 Nov 2025).

For classical groups of types B, C, D, the tensor product formula involves composite horizontal-strip configurations: $$\Pi_\lambda\otimes\wedge^i V = \bigoplus_{\mu} \Pi_\mu,$$ where $\mu$ is obtained by a two-step process: $\lambda\subset\xi\supset\mu$, with $\xi/\lambda$ and $\xi/\mu$ horizontal strips and $|\xi|-|\lambda| + |\xi|-|\mu|=i$ (or $i$ or $i-1$ in type B) (Biswas, 21 Dec 2025).

2. Combinatorial Models and Tableau Rules

The heart of Pieri rules for classical groups is combinatorial: the selection of valid "moves"—adding strips or rims to Young diagrams under stringent conditions. Key models include:

• Horizontal and Vertical Strips: For $GL_n$, a horizontal $r$-strip adds $r$ Boxes with no two in the same column, while a vertical $r$-strip prohibits two Boxes in the same row. These underpin the classical expansions and are reflected in Littlewood-Richardson tableaux and the RSK correspondence (Biswas, 21 Dec 2025).
• Generalized Tableaux: In type C—Lagrangian Grassmannian topology or $K$-theory—the valid moves are described via KLG-tableaux (strictly increasing with unprimed/primed labels and specific diagonal constraints). For type D—maximal orthogonal Grassmannian—KOG-tableaux provide the analogous rule, replacing symmetry/priming with parity of entries and diagonal restrictions (Buch et al., 2010, Arroyo, 7 Nov 2025).
• Affinization and Noncommutative Extensions: The affine Grassmannian context introduces affine Stanley symmetric functions and noncommutative $k$-Schur functions. The multiplication rules (Pieri rules) here involve "Pieri factors" in the affine nil-Coxeter algebra, controlled by the statistics $\mathrm{stat}(v)$ and support combinatorics in the Coxeter group (Pon, 2011, Lam et al., 2011).
• Insertion/Bumping Algorithms: Multiplicities in the combinatorial expansions can be interpreted algorithmically—e.g., through Berele's symplectic insertion and its orthosymplectic generalization, or as bijections between classes of tableaux and bumping sequences. The orthosymplectic Pieri rule (which coincides with Sundaram's symplectic case) counts chains of partitions with successive horizontal strips, parameterizing the admissible insertions (Stokke, 2018).

3. Branching Laws and Reciprocity

Pieri rules are intimately tied to branching laws when restricting representations from a group $G$ to a subgroup $H$. For instance:

• GL(n+1) to GL(n) Branching: The classic "interlacing" branching corresponds bijectively to the horizontal-strips Pieri rule for $GL_n$: a $\lambda$ appears in $\mu|_{GL_n}$ iff $\lambda\subset\mu$ as a horizontal strip (Biswas, 21 Dec 2025, Rajan et al., 2023).
• Relative Pieri Formulas: The setting of Rajan–Shrivastava (Rajan et al., 2023) gives explicit "relative Pieri" formulas for pairs $(G, H)$ such as $(GL(n+1), GL(n))$, $(Spin(2n+1), Spin(2n))$, and $(Sp(2n), Sp(2)\times Sp(2n-2))$. For example, the relative Pieri formula for $(GL(n+1), GL(n))$ expresses the product $\chi_\mu \cdot \Delta$ as a sum over all ways to add $0$ or $1$ boxes to each row (subject to dominance), with the sign tracking the number of non-added rows.
• Duality and Reciprocity Laws: In the symplectic case, the skew Pieri rule follows from a deeper skew-duality (Howe) and a reciprocity between tensor product decompositions and branching multiplicities, where one side interprets Pieri-like expansions as branching to products of smaller symplectic groups (Howe et al., 2016).

4. Geometric and Equivariant Formulations

Geometry provides alternate frameworks for Pieri rules, especially through the topology of homogeneous spaces and flag varieties:

• Schubert Calculus: The multiplication of Schubert cycles (in cohomology, $K$-theory, or quantum cohomology) is governed by Pieri-type rules localized to special subvarieties or implemented through push-forward and pull-back in equivariant cohomology (Li et al., 2018, Leung et al., 2013).
• Equivariant and Quantum Extensions:
  • Equivariant Pieri rules supply explicit positivity formulas for products of torus-invariant classes, with coefficients as polynomials in (difference of) weights.
  • Quantum Pieri rules introduce quantum parameters, where corrections to the classical expansions arise from genus-0 Gromov–Witten invariants (computable by recursion or via degenerations to smaller Grassmannians in types B, C, D) (Leung et al., 2013).
• K-theory: Pieri rules in $K$-theory (for both type A and isotropic Grassmannians) involve signed counts of tableau fillings and recursive formulas for rim-shaped length modifications to the Young diagram, with combinatorial tableau models such as set-valued tableaux, KOG-tableaux, KLG-tableaux, and strict decomposition tableaux (Buch et al., 2010, Arroyo, 7 Nov 2025).

5. Extended Kostant Theorem and Levi Subgroups

The multiplicity-freeness and structure of the classical Pieri rules are explained by the relation to extended forms of Kostant's tensor product theorem and the action of Levi subgroups:

• Extended Kostant Bound: If $\Pi_\nu\subset\Pi_\lambda\otimes\Pi_\mu$, then $\nu=\lambda+\tilde{\mu}$ with $\tilde{\mu}$ a weight of $\Pi_\mu$ and the multiplicity bounded accordingly. The converse for appropriate minuscule modules is, in fact, equivalent to the Pieri rule (Biswas, 21 Dec 2025).
• Levi Subgroup Perspective: For Siegel and related parabolics with Levi $GL_n$, Pieri rules and branching laws can be viewed as consequences of constraints at the $GL_n$ level, so the allowed diagrams and strip additions in $G$ correspond to those obtainable via $GL_n$-theoretic Pieri moves (Biswas, 21 Dec 2025).

6. Representative Examples and Structural Table

Below is a table summarizing the combinatorial structure of the classical and quantum Pieri rules across types.

Group / Type	Combinatorial Condition	Model / Tableaux
$GL_n$ (Type A)	Horizontal/vertical strip	Semistandard tableaux (RSK)
$Sp_{2n}$ (Type C)	Two horizontal strips, sum $=i$	Symplectic/Berele insertion
$SO_{2n},\,SO_{2n+1}$	Two horizontal strips, sum $=i$ or $i-1$	Sundaram insertion, KOG/KLG
Affine (all types)	Cyclically-decreasing elements, Pieri factors in nil-Coxeter	Grassmannian elements, k-Schur functions
$K$-theory	Rims, arms, sign rules	Set-valued, strict, KOG, KLG, SDT tableaux

Each case is furnished with an explicit combinatorial recipe and, wherever quantum or $K$-theoretic corrections occur, recursion or tableau enumeration is provided.

7. Implications and Extensions

The unifying role of Pieri rules emerges across several domains:

• Representation Theory: They govern branching, tensor products, and the structure of highest weight modules for classical groups over both $\mathbb{C}$ and finite fields (Gurevich et al., 2021).
• Geometry and Schubert Calculus: They underpin algorithms for intersection numbers, equivariant and quantum multiplicative structures on (co)homology and $K$-theory.
• Symmetric Function Theory and Combinatorics: Pieri formulas manifest as expansion rules for Schur, Schubert, and related functions (keys, affine Stanley symmetric), driving developments in symmetric function generalizations, Demazure character theory, and non-commutative symmetric function literature (Assaf et al., 2019, Pon, 2011).
• Reciprocity and Duality: The connection to branching demonstrates an explicit duality between induction/restriction in representation theory and geometric transition (box-adding vs. box-removal), further illustrated via Howe duality and relative Pieri formulas (Rajan et al., 2023, Biswas, 21 Dec 2025).

This rich interconnection of combinatorics, geometry, and representation theory ensures that any new approach or explicit model for the Pieri rules—such as strict decomposition tableaux, insertions, or Levi-theoretic induction—yields not just proof techniques but direct computational and structural insight across the landscape of classical groups.