geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
Young diagrams (also called Ferrers diagrams) are used to describe many objects in algebra and combinatorics, including:
integer partitions. For example, the integer partition
is drawn as the Young diagram
conjugacy classes in $S_n$.
irreducible representations of the symmetric groups $S_n$ over any field of characteristic zero
irreducible (algebraic) representation of the special linear groups $SL(N,\mathbb{C})$
irreducibleunitary representations of the special unitary groups $SU(N)$
basis vectors for the free lambda-ring on one generator, $\Lambda$
flag varieties for the special linear groups $SL(N,k)$, where $k$ is any field
characteristic classes for complex vector bundles: that is, cohomology classes on the classifying spaces of the general linear groups $GL(N,\mathbb{C})$
characteristic classes for hemitian vector bundles: that is, cohomology classes on the classifying spaces $B U(N)$ of the unitary groups $U(N)$
finite-dimensional C*-algebras: any such algebra is of the form $M_{n_1}(\mathbb{C}) \oplus \cdots \oplus M_{n_k}(\mathbb{C})$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.
finite abelian p-groups: any such group is of the form $\mathbb{Z}/p^{n_1} \oplus \cdots \oplus \mathbb{Z}/p^{n_k}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.
finite commutative semisimple algebras over a prime field $\mathbb{F}_p$: any such algebra is of the form $\mathbb{F}_{p^{n_1}} \oplus \cdots \oplus \mathbb{F}_{p^{n_k}}$ for some unique list of natural numbers $n_1 \ge n_2 \ge \cdots \ge n_k$.
the trace of the category of finite sets has isomorphism classes of objects corresponding to Young diagrams.
A Young diagram $F^\lambda$, also called Ferrers diagram, is a graphical representation of an unordered integer partition $\lambda = (\lambda_1\ge\lambda_2\ge\ldots\ge\lambda_l$). If $\lambda\vdash n$ is a partition of $n$ then the Young diagram has $n$ boxes. A partition can be addressed as a multiset over $\mathbb{N}$.
There are two widely used such representations. The English one uses matrix-like indices, and the French one uses Cartesian coordinate-like indices for the boxes $x_{i,j}$ in the diagram $F^\lambda$.
In the English representation the boxes are adjusted to the north-west in the 4th quadrant of a 2-dimensional Cartesian coordinate system, with the ‘y’-axis being downward oriented. For instance the diagram $F^{(5,4,4,2,1,1)}$ representing the partition $(5,4,4,2,1,1)$ of $17$ is given in the English representation as:
Let $\mathbb{Y}$ be the set of Young diagrams. Important functions on Young diagrams include:
A filling of a Young diagram with elements from a set $S$ is called a Young tableau.
A generalization of a Young diagram is a skew Young diagram. Let $\mu,\nu$ be two partitions, and let $\nu \le \mu$ be defined as $\forall i : \nu_i\le \mu_i$ (possibly adding trailing zeros). The skew Young diagram $F^{\mu/\nu}$ is given by the Young diagram $F^\mu$ with all boxes belonging to $F^\nu$ when superimposed removed. If $\mu=(5,4,4,2,1,1)$ and $\nu=(3,3,2,1)$ then $F^{\mu/\nu}$ looks like:
…
Quick introduction
Textbook accounts are in any book on representation theory in general and on the representation theory of the symmetric group in particular; such as:
William Fulton, Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991 (pdf)
Bruce E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer, 2001
More details:
Kazuhiko Koike, Itaru Terada, Young-diagrammatic methods for the representation theory of the classical groups of type $B_n$, $C_n$, $D_n$, Journal of Algebra, Volume 107, Issue 2, May 1987, Pages 466-511
William Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997 (doi:10.1017/CBO9780511626241)
Ron M. Adin, Yuval Roichman, Enumeration of Standard Young Tableaux, Chapter 14 in: Miklós Bóna, Handbook of Enumerative Combinatorics, CRC Press 2015 (arXiv:1408.4497, ISBN:9781482220858)
Connection to algebraic geometry:
With an eye towards application to (the standard model of) particle physics:
Last revised on July 10, 2024 at 13:15:36. See the history of this page for a list of all contributions to it.