Schur algebra

The Schur algebra ${\displaystyle S_{k}(n,r)}$ can be defined for any commutative ring ${\displaystyle k}$ and integers ${\displaystyle n,r\geq 0}$. Consider the algebra ${\displaystyle k[x_{ij}]}$ of polynomials (with coefficients in ${\displaystyle k}$) in ${\displaystyle n^{2}}$ commuting variables ${\displaystyle x_{ij}}$, 1 ≤ i, j ≤ ${\displaystyle n}$. Denote by ${\displaystyle A_{k}(n,r)}$ the homogeneous polynomials of degree ${\displaystyle r}$. Elements of ${\displaystyle A_{k}(n,r)}$ are k-linear combinations of monomials formed by multiplying together ${\displaystyle r}$ of the generators ${\displaystyle x_{ij}}$ (allowing repetition). Thus

${\displaystyle k[x_{ij}]=\bigoplus _{r\geq 0}A_{k}(n,r).}$

Now, ${\displaystyle k[x_{ij}]}$ has a natural coalgebra structure with comultiplication ${\displaystyle \Delta }$ and counit ${\displaystyle \varepsilon }$ the algebra homomorphisms given on generators by

${\displaystyle \Delta (x_{ij})=\textstyle \sum _{l}x_{il}\otimes x_{lj},\quad \varepsilon (x_{ij})=\delta _{ij}\quad }$    (Kronecker's delta).

Since comultiplication is an algebra homomorphism, ${\displaystyle k[x_{ij}]}$ is a bialgebra. One easily checks that ${\displaystyle A_{k}(n,r)}$ is a subcoalgebra of the bialgebra ${\displaystyle k[x_{ij}]}$, for every r ≥ 0.

Definition. The Schur algebra (in degree ${\displaystyle r}$) is the algebra ${\displaystyle S_{k}(n,r)=\mathrm {Hom} _{k}(A_{k}(n,r),k)}$. That is, ${\displaystyle S_{k}(n,r)}$ is the linear dual of ${\displaystyle A_{k}(n,r)}$.

It is a general fact that the linear dual of a coalgebra ${\displaystyle A}$ is an algebra in a natural way, where the multiplication in the algebra is induced by dualizing the comultiplication in the coalgebra. To see this, let

${\displaystyle \Delta (a)=\textstyle \sum a_{i}\otimes b_{i}}$

and, given linear functionals ${\displaystyle f}$, ${\displaystyle g}$ on ${\displaystyle A}$, define their product to be the linear functional given by

${\displaystyle \textstyle a\mapsto \sum f(a_{i})g(b_{i}).}$

The identity element for this multiplication of functionals is the counit in ${\displaystyle A}$.

• One of the most basic properties expresses ${\displaystyle S_{k}(n,r)}$ as a centralizer algebra. Let ${\displaystyle V=k^{n}}$ be the space of rank ${\displaystyle n}$ column vectors over ${\displaystyle k}$, and form the tensor power

${\displaystyle V^{\otimes r}=V\otimes \cdots \otimes V\quad (r{\text{ factors}}).}$

Then the symmetric group ${\displaystyle {\mathfrak {S}}_{r}}$ on ${\displaystyle r}$ letters acts naturally on the tensor space by place permutation, and one has an isomorphism

${\displaystyle S_{k}(n,r)\cong \mathrm {End} _{{\mathfrak {S}}_{r}}(V^{\otimes r}).}$

In other words, ${\displaystyle S_{k}(n,r)}$ may be viewed as the algebra of endomorphisms of tensor space commuting with the action of the symmetric group.

• ${\displaystyle S_{k}(n,r)}$ is free over ${\displaystyle k}$ of rank given by the binomial coefficient ${\displaystyle {\tbinom {n^{2}+r-1}{r}}}$.
• Various bases of ${\displaystyle S_{k}(n,r)}$ are known, many of which are indexed by pairs of semistandard Young tableaux of shape ${\displaystyle \lambda }$, as ${\displaystyle \lambda }$ varies over the set of partitions of ${\displaystyle r}$ into no more than ${\displaystyle n}$ parts.
• In case k is an infinite field, ${\displaystyle S_{k}(n,r)}$ may also be identified with the enveloping algebra (in the sense of H. Weyl) for the action of the general linear group ${\displaystyle \mathrm {GL} _{n}(k)}$ acting on ${\displaystyle V^{\otimes r}}$ (via the diagonal action on tensors, induced from the natural action of ${\displaystyle \mathrm {GL} _{n}(k)}$ on ${\displaystyle V=k^{n}}$ given by matrix multiplication).
• Schur algebras are "defined over the integers". This means that they satisfy the following change of scalars property:

${\displaystyle S_{k}(n,r)\cong S_{\mathbb {Z} }(n,r)\otimes _{\mathbb {Z} }k}$

for any commutative ring ${\displaystyle k}$.

• Schur algebras provide natural examples of quasihereditary algebras[4] (as defined by Cline, Parshall, and Scott), and thus have nice homological properties. In particular, Schur algebras have finite global dimension.

• Generalized Schur algebras (associated to any reductive algebraic group) were introduced by Donkin in the 1980s.[5] These are also quasihereditary.
• Around the same time, Dipper and James[6] introduced the quantized Schur algebras (or q-Schur algebras for short), which are a type of q-deformation of the classical Schur algebras described above, in which the symmetric group is replaced by the corresponding Hecke algebra and the general linear group by an appropriate quantum group.
• There are also generalized q-Schur algebras, which are obtained by generalizing the work of Dipper and James in the same way that Donkin generalized the classical Schur algebras.[7]
• There are further generalizations, such as the affine q-Schur algebras[8] related to affine Kac–Moody Lie algebras and other generalizations, such as the cyclotomic q-Schur algebras[9] related to Ariki-Koike algebras (which are q-deformations of certain complex reflection groups).

The study of these various classes of generalizations forms an active area of contemporary research.

• Stuart Martin, Schur Algebras and Representation Theory, Cambridge University Press 1993. MR2482481, ISBN 978-0-521-10046-5
• Andrew Mathas, Iwahori-Hecke algebras and Schur algebras of the symmetric group, University Lecture Series, vol.15, American Mathematical Society, 1999. MR1711316, ISBN 0-8218-1926-7
• Hermann Weyl, The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton, N.J., 1939. MR0000255, ISBN 0-691-05756-7