SchurFunctors -- Schur and Weyl functors from partitions and Young tableaux

A Schur functor $S_\lambda$ is a polynomial functor on the category of free modules, indexed by a partition $\lambda = (\lambda_1 \ge \lambda_2 \ge \cdots \ge \lambda_r)$. Classical special cases include

• $S_{(d)}(E) = \mathrm{Sym}^d(E)$, the $d$-th symmetric power
• $S_{(1^d)}(E) = \wedge^d(E)$, the $d$-th exterior power
• $S_{(d,\dots,d)}(E) = (\det E)^{\otimes k}$ on a rank-$d$ factor (repeated $k$ times)
• $S_{(2,1)}(E) \subseteq E \otimes \wedge^2 E$, the first genuinely mixed case

The Weyl functor $W_\lambda$ is the divided-power analogue: in characteristic zero $W_\lambda(E) \cong S_\lambda(E)$, but over $\mathbb{Z}$ or in positive characteristic they are genuinely different functors. They are related by duality and appear symmetrically in Schur-Weyl duality and in the definition of Weyl modules of the general linear group.

Construction. Both functors are presented by Garnir / straightening relations. For $S_\lambda$ one has the presentation $$\textstyle\bigotimes_{i \ge 1}\wedge^{\lambda'_i}(E) \;\twoheadrightarrow\; S_\lambda(E),$$ where $\lambda'$ is the conjugate (transpose) partition: $S_\lambda(E)$ is the quotient of this tensor product of exterior powers by the Garnir relations across adjacent columns. Dually, $W_\lambda$ is a quotient of a tensor product of divided powers, $$\textstyle\bigotimes_{i \ge 1} D^{\lambda_i}(E) \;\twoheadrightarrow\; W_\lambda(E),$$ by (divided-power) Garnir relations across adjacent rows.

Tableaux and the semistandard basis. A filling of a Young diagram of shape $\lambda$ by integers $\{0,\dots,n-1\}$ (representing a basis of $E = R^n$) is semistandard if its entries weakly increase along rows and strictly increase down columns. Over $\mathbb{Z}$ the semistandard fillings index a free basis of $S_\lambda(E)$ and of $W_\lambda(E)$. Arbitrary fillings are reduced to combinations of semistandard ones by the straightening algorithm; see straighten and weylStraighten.

In this package Schur tableaux are stored column-wise as objects of type Filling, while Weyl tableaux are stored row-wise as objects of type WeylFilling. Conjugation of partitions swaps the two conventions; see conjugate(Filling).

Representation theory. Over a field of characteristic zero the Schur functors $S_\lambda(V)$, as $\lambda$ ranges over partitions with at most $\dim V$ parts, give all the polynomial irreducible representations of $GL(V)$. The character of $S_\lambda(V)$ is the Schur function $s_\lambda$ in the eigenvalues of the diagonal torus; computing and decomposing characters is handled by character, weylCharacter, splitCharacter, characterRep, and decomposeRep.

Package layout. The two worlds (Schur / Weyl) expose parallel sets of functions:

• Modules: schurModule / weylModule.
• Functoriality: schur / weyl take a map $f : E \to F$ to the induced $S_\lambda(f)$ / $W_\lambda(f)$.
• Straightening: straighten / weylStraighten rewrite an arbitrary tableau in the semistandard basis.
• Module maps: schurModulesMap / weylModulesMap build a homomorphism from its action on a tableau basis.
• Tableau combinatorics: Filling / WeylFilling, with normalize / weylNormalize, isStandard / isWeylStandard, towardWeylStandard, standardTableaux / standardWeylTableaux, augmentFilling / augmentWeylFilling, maxWeylFilling.
• Divided powers: dividedPower, divMult, divComult.
• Characters: character / weylCharacter, splitCharacter, characterRep, decomposeRep.
• Pretty-printing: printSchurModuleElement / printWeylModuleElement.

A first example. Take the smallest non-hook, non-trivial shape $(2,1)$:

Both have rank 8, which is $\dim S_{(2,1)}(\mathbb{Q}^3) = 8$ by the Weyl character formula. They are canonically isomorphic in characteristic 0, but are presented as quotients of different ambients: $\wedge^2(E)\otimes E \twoheadrightarrow \mathrm{schurModule}((2,1), \mathbb{Q}^3)$ while $D^2(E)\otimes D^1(E) \twoheadrightarrow \mathrm{weylModule}((2,1), \mathbb{Q}^3)$.

References. For the Schur side see W. Fulton, Young Tableaux, LMS Student Texts 35 (1997), Chapter 8. For the Weyl-module / divided-power construction see K. Akin, D. Buchsbaum, and J. Weyman, Schur functors and Schur complexes, Adv. Math. 44 (1982), 207-278; and the book J. Weyman, Cohomology of Vector Bundles and Syzygies, Cambridge Tracts 149 (2003).