Glossary¶
Coxeter Groups¶
- Cartan matrix
A matrix \(C = (c_{ij})\), with \(1 \leqslant i,j \leqslant n\) and \(c_{ij} \in \mathbb{R}\), such that:
- for \(i\neq j\) we have \(c_{ij} \leqslant 0\)
- \(c_{ij} \neq 0\) if and only if \(c_{ji} \neq 0\)
- \(c_{ii} = 2\)
- for \(i \neq j\) we have \(c_{ij}c_{ji} = 4\cos^2(\pi/m_{ij})\) where \(m_{ij} \geqslant 2\) or \(m_{ij} = \infty\).
- Coxeter group
- A group \(W\) containing a generating set \(\mathbb{S}\) such that \((W,\mathbb{S})\) is a Coxeter system.
- Coxeter matrix
A symmetric matrix \(M = (m_{ij})\), with \(1 \leqslant i,j \leqslant n\) and \(m_{ij} \in \mathbb{N} \cup \{\infty\}\), such that
- \(m_{ii} = 1\) for all \(i\),
- \(m_{ij} > 1\) for all \(i \neq j\).
- Coxeter system
- A pair \((W,\mathbb{S})\) such that \(W\) is a group and \(\mathbb{S} \subset W\) is a finite generating set such that the defining relations for \(W\) are just relations of the form \(s^2 = 1\) and \((st)^{m_{st}} = 1\), where \(s \neq t \in \mathbb{S}\) and \(m_{st} = m_{ts} > 1\). If, for two generators \(s,t \in \mathbb{S}\) there exists no \(i \in \mathbb{N}\) such that \((st)^i = 1\) then as a convention we set \(m_{st} = \infty\).
- length function
- For a Coxeter system \((W,\mathbb{S})\) this is the function \(\ell : W \to \mathbb{N}\) defined by setting \(\ell(1) = 0\) and \(\ell(w) = r\) if \(s_1\cdots s_r\) is a reduced expression for \(w \neq 1\).
- reduced expression
- Given a Coxeter system \((W,\mathbb{S})\) and an element \(w \in W\) this is a word \(w = s_1\cdots s_r\) in the generators \(\mathbb{S}\) such that any other word \(w = s_1\cdots s_{r'}\) satisfies \(r' \geqslant r\).
Representation Theory¶
- generic degree polynomial
- Assume \(W\) is a finite Coxeter group and \(V\) is the natural module for \(W\).
- j-induction
For any coxeter group \(W\) and subgroup \(H\) this is the following map \(j_H^W : \mathrm{Cent}(H) \to \mathrm{Cent}(W)\) taking class functions on \(H\) to class functions on \(W\). Denote by \(n_{\chi,\rho}\) the multiplicity of \(\rho \in \mathrm{Irr}(W)\) in the usual induced character \(\mathrm{Ind}_H^W(\chi)\). Then
\[j_H^W(\chi) = \sum_{\rho} n_{\chi,\rho} \rho\]where the sum is taken over all \(\rho \in \mathrm{Irr}(W)\) such that \(b_{\rho} = b_{\chi}\), where \(b_{\chi}\) denotes the b-invariant of \(\chi\). See 5.2 of [GP00] for a generalisation of this to all finite groups.
- preferred extension
- Assume \((W,\mathbb{S})\) is a Coxeter system such that \(W\) is a Weyl group and \(\phi : W \to W\) is an automorphism stabilising \(\mathbb{S}\). Every \(\phi\)-invariant irreducible character of \(W\) may be extended to an irreducible character of the semidirect product \(W \rtimes \langle \phi \rangle\). This term refers to the extension defined by Lusztig in 17.2 of [Lus85].
- Schur element
If \(\chi\) is an irreducible character of an Iwahori-Hecke algebra \(H_{\mathcal{A}}(W,\mathbb{S},\{a_s,b_s \mid s \in \mathbb{S}\})\) then the associated Schur element \(c_{\chi}\) is the unique element for which the following relation holds
\[\begin{split}\sum_{\chi} \chi(T_w)c_{\chi}^{-1} = \begin{cases} 1 &\text{if }w=1,\\ 0 &\text{if }w\neq 1, \end{cases}\end{split}\]where the sum runs over all the irreducible characters of the algebra.
- sign character
- Given a Coxeter system \((W,\mathbb{S})\) this is the homomorphism \(\mathrm{sgn} : W \to \mathbb{C}^{\times}\) obtained by setting \(\mathrm{sgn}(w) = (-1)^{\ell(w)}\) where \(\ell : W \to \mathbb{N}\) is the length function.
Hecke Algebras¶
- Iwahori-Hecke algebra
Fix a Coxeter system \((W,\mathbb{S})\) and a commutative ring \(\mathcal{A}\) with 1. Furthermore let us assume that \(\{a_s,b_s \mid s \in \mathbb{S}\} \subset \mathcal{A}\) are such that \(a_s = a_t\) and \(b_s = b_t\) whenever \(s,t \in \mathbb{S}\) are conjugate in \(W\). The Iwahori-Hecke algebra \(H_{\mathcal{A}}(W,\mathbb{S},\{a_s,b_s \mid s \in \mathbb{S}\})\) is the \(\mathcal{A}\)-algebra generated by \(\{T_s \mid s \in \mathbb{S}\}\) and satisfying the relations
\[\begin{split}\begin{cases} T_s^2 = a_sT_1 + b_sT_s &\text{for }s\in \mathbb{S}\\ T_sT_tT_s \cdots = T_tT_sT_t \cdots &\text{for }s\neq t\text{ and with }m_{st}\text{ factors on each side}. \end{cases}\end{split}\]Here \(m_{st}\) is as in the definition of Coxeter system. This algebra is in fact free as an \(\mathcal{A}\)-module with basis given by \(\{T_w \mid w \in W\}\). It also satisfies the multiplication relation
\[\begin{split}T_sT_w = \begin{cases} T_{sw} &\text{if }\ell(sw) > \ell(w)\\ a_sT_{sw} + b_sT_w &\text{if }\ell(sw) < \ell(w) \end{cases}\end{split}\]for all \(s \in \mathbb{S}\) and \(w \in W\), where \(\ell\) is the length function of \((W,\mathbb{S})\).
- generic Iwahori-Hecke algebra
- Fix a Coxeter system \((W,\mathbb{S})\) and indeterminates \(\{u_s \mid s \in \mathbb{S}\}\) over \(\mathbb{C}\) such that \(u_s = u_t\) whenever \(s,t \in \mathbb{S}\) are conjugate in \(W\). Let \(\mathcal{A} = \mathbb{Z}[u_s^{\pm} \mid s \in \mathbb{S}]\) and assume that there exists a ring homomorphism \(\mathcal{A} \to \mathbb{Z}\) which sends \(u_s \to 1\) for all \(s \in \mathbb{S}\). Then the generic Iwahori-Hecke algebra is defined to be the Iwahori-Hecke algebra \(H_{\mathcal{A}}(W,\mathbb{S},\{u_s,u_s-1 \mid s \in \mathbb{S}\})\).
Combinatorics¶
- bipartition
- A pair of ordered partitions.
- partition
- A finite weakly decreasing sequence of non-zero natural numbers.
- dual partition
If \(\lambda = (\lambda_1,\dots,\lambda_r)\) is a partition of \(n\) then the dual partition is the partition of \(n\) obtained in the following way. First let \(\lambda' = (\lambda_1',\dots,\lambda_n')\) be the sequence obtained by setting
\[\lambda_i' = |\{\lambda_j \mid 1 \leqslant j \leqslant n \text{ and }\lambda_j \geqslant i\}|.\]Then \(\lambda^*\) is obtained from \(\lambda'\) by removing all entries equal to zero.