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For a permutation $\pi$ of $\{1,\dots ,n\}$, an inversion is a pair $i<j$ with $\pi (i)>\pi (j)$. Its number equals the Coxeter length for adjacent transpositions: each adjacent swap changes the count by one, and swapping adjacent descents sorts the permutation in exactly that many steps.

For a chosen positive system of a root system $\Pi$, the inversion set isFor a finite Weyl group, $|N(w)|=\ell (w)$.

Its cardinality is the Coxeter length; this root-sign definition should not be silently identified with an unspecified permutation inversion convention.

A group length function $L:G\to [0,\infty )$ satisfies $L(1)=0$, $L(g^{-1})=L(g)$ and $L(gh)\le L(g)+L(h)$. Word length for a chosen generating set is an example; Coxeter length uses the simple reflections.

The longest element $w_0$ is the unique element of maximal Coxeter length in a finite Coxeter group. It is an involution and satisfies $\ell (w_{0}w)=\ell (w_0)-\ell (w)$.

Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra, choose a Cartan subalgebra $\mathfrak{t}$ and a set $R^{+}$ of positive roots, and write $W$ for the Weyl group. Let $\lambda$ be a dominant integral weight and $L(\lambda )$ the finite-dimensional irreducible highest-weight representation with that highest weight. If $L(\lambda {)}_{\mu }$ denotes its weight space of weight $\mu$, its formal character is $\mathrm{ch}L(\lambda )={\sum }_{\mu }\dim L(\lambda {)}_{\mu }{e}^{\mu }$. The symbols $e^{\mu }$ belong to the group algebra of the weight lattice and multiply by $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$; they are formal symbols, not matrix exponentials.

Put $\rho =\frac{1}{2}{\sum }_{\alpha \in R^{+}}\alpha$, the half-sum of positive roots, and let $\ell (w)$ be the Coxeter length of $w$, the least number of simple reflections expressing it. Then $(-1{)}^{\ell (w)}=\det (w)$ in the real reflection representation. The Weyl character formula statesBy the Weyl denominator formula, the denominator is $e^{\rho }{\prod }_{\alpha \in R^{+}}(1-e^{-\alpha })$, giving the equivalent expressionThe quotient can be interpreted in the fraction field of the formal group algebra; the theorem says it simplifies to the finite formal character above. In particular the apparent denominator does not mean that the representation has infinitely many weights.

The Weyl group acts on the weight lattice by its root reflections and extends to ring automorphisms of the group ring of a weight lattice:Weights and their multiplicities in a finite-dimensional Lie algebra representation of a semisimple Lie algebra are permuted by this action, so its formal character is Weyl-invariant.

Choose a set ${\Phi }^{+}$ of positive roots and putHere ${\omega }_i$ are the fundamental weights, and $\det (w)=(-1{)}^{\ell (w)}$, where the Coxeter length $\ell (w)$ counts positive roots carried to negative ones. For a dominant integral weight $\lambda$, let $L(\lambda )$ denote the finite-dimensional Irreducible Lie algebra representation of highest weight $\lambda$. The Weyl character formula isThe second denominator expression is the Weyl denominator formula. Both alternants change sign under each root reflection; their quotient is invariant. Although written as a fraction, the theorem asserts that it is a finite element of $\mathbb{Z}[{\Lambda }_W]$ with the nonnegative coefficients given by weight multiplicities. The highest-weight classification specifies which $\lambda$ label the irreducibles, while the root system and choice of positive roots determine every other term.

Use a Gelfand–Tsetlin basis in Young seminormal form.

For a standard Young tableau $T$, put $d=c_T(i+1)-c_T(i)$, using the Content of a Young-diagram cell. Choose the row-reading tableau $T_0$, and let $\ell (T)$ be the Coxeter length of the unique permutation sending $T_0$ to $T$. A Gelfand–Tsetlin basis can be chosen so that, when $R=s_{i}T$ is standard and $\ell (R)=\ell (T)+1$,If $R$ is not standard, the action is $+v_T$ for two consecutive entries in one row, and $-v_T$ for two in one column. This is one usual normalization of the Young seminormal form.

Here is a construction and proof of the normalization. Fix $v_{T_0}\ne 0$, let $P_T$ be the projection onto the tableau line, and defineThe permutation ${\pi }_T$ has a reduced expression consisting entirely of admissible swaps, by the reduced adjacent-swap path between linear extensions. At each swap the off-diagonal coefficient is nonzero. In its expansion, the only term that can reach a tableau at distance $\ell (T)$ uses all $\ell (T)$ swaps; omitting a swap gives a shorter path. Thus $v_T\ne 0$. This also makes its definition independent of a chosen reduced expression, because ${\pi }_T$ itself is fixed.

The relation $s_i{X}_i{s}_i+s_i=X_{i+1}$ forces the coefficient of $v_T$ in $s_i{v}_T$ to be $1/d$, and forces every other component to lie on the swapped line. If length increases, project the identity ${\pi }_R=s_i{\pi }_T$ onto the line for $R$. The shorter terms of ${\pi }_T{v}_{T_0}$ cannot reach $R$ in one step, so the coefficient of $v_R$ is exactly one. Applying $s_i^2=1$ then gives the reverse coefficient $1-d^{-2}$ and diagonal coefficient $-1/d$. In the nonstandard cases the same relation gives $d=\pm 1$ and the asserted scalar action. This proves the theorem rather than merely specifying pairwise scalings.

The inclusion here is inclusion of Young diagrams, not the dominance order on partitions. The skew Young diagram $\lambda /\mu$ is the set difference of their cells. Two cells are adjacent when they share an edge. It is connected when any two cells can be joined by such steps, and is a rim hook when it is connected and contains no $2\times 2$ square. Under the usual edge-adjacency convention a totally disconnected skew Young diagram has only singleton components, equivalently no two cells share an edge. A horizontal strip instead means at most one cell per column; this distinction matters for the last part of this question.

A standard skew Young tableau is a linear extension of a partially ordered set: the cells are ordered by the row and column inequalities, and the tableau lists them in increasing label order. Write the current list as $x_1,\dots ,x_k$ and let $r_j$ be the desired label of $x_j$ in $R$. Then $(r_1,\dots ,r_k)$ is the one-line notation of the unique permutation $\pi$ with $\pi T=R$.

Whenever this list of desired labels is not increasing, there is an adjacent descent $r_j>r_{j+1}$. The cells $x_j,x_{j+1}$ are incomparable in the cell partial order: if they were comparable, both $T$ and $R$ would have to put them in the same order. Interchanging their consecutive current labels is therefore admissible. It removes exactly one inversion of a permutation from the list of desired labels. Repeating ends at $R$ after exactly the original inversion count, which is the Coxeter length $\ell (\pi )$. ThusThis proves the reduced adjacent-swap path between linear extensions and applies equally to ordinary tableaux.

Order the chambers $wK$ by nondecreasing Coxeter length, beginning with $K$. When $wK$ is attached, letbe its right descent set. Claim C2 applied to the coset $w{W}_{T(w)}$, followed by C1, shows that $T(w)$ is spherical. The part of $wK$ already present is exactlyIt is nonempty for $w\ne e$ and is contractible by C3. The chamber $wK$ is contractible as well, so C4 shows inductively that every finite length-ordered union of chambers is contractible.

The Davis complex is a CW complex and is the increasing union of these chamber unions. Every map from a sphere has compact image and therefore lies in a finite union; the next finite contractible union null-homotopes it. Thus every homotopy group of the Davis complex vanishes. Since it is connected, the Whitehead theorem implies

For a dominant integral weight $\lambda$, the Weyl character formula isHere $W$ is the Weyl group, $\ell (w)$ its Coxeter length, $\rho$ the Weyl vector, and $\mathrm{ch}$ the formal character of a weight module. Taking the limit at the identity gives the Weyl dimension formula

For a dominant integral weight $\lambda$, the Weyl character formula iswhere $W$ is the Weyl group, $\ell$ is Coxeter length, and $\rho$ is the half-sum of positive roots. The Weyl denominator formula is

Set $\lambda =k\rho$. Apply the denominator identity after replacing every formal exponential $e^{\mu }$ by $e^{(k+1)\mu }$:Dividing this by the ordinary denominator gives

We induct on the Coxeter length $\ell (w)$. There is nothing to prove when $\ell (w)=0$. Otherwise choose a simple root $\alpha$ such thatequivalently, $w\alpha$ is negative. Since $\lambda$ and $\mu =w\lambda$ lie in the Closed dominant Weyl chamber,Therefore $⟨\lambda ,{\alpha }^{\vee }⟩=0$, and the simple reflection $s_{\alpha }$ fixes $\lambda$. Moreover,The induction hypothesis writes $w{s}_{\alpha }$ as a product of simple reflections that fix $\lambda$. Multiplying on the right by $s_{\alpha }$ gives the required expression for $w$. This is the Weyl stabilizer of a dominant point lemma.

If a finitely generated Coxeter group $W$ is finite, its integer-valued Coxeter length has a maximum. Conversely, if some $w_0$ has globally maximal length $N$, every group element has a word of length at most $N$. There are only finitely many words of bounded length in the finite set of simple generators, so $W$ is finite.

Realize the finite group as the reflection group of a root system with fundamental system $\Delta$ and positive system $\Pi$. Maximality and the fact that multiplication by a simple generator changes Coxeter length by one giveThe positive-root criterion for Coxeter length therefore gives $w_0\Delta \subseteq -\Pi$. Since $w_0\Delta$ is itself fundamental, it must be the simple system $-\Delta$ of the positive system $-\Pi$.

If $v_0$ is another maximal-length element, the same argument gives $v_0\Delta =-\Delta =w_0\Delta$. Hence $v_0^{-1}{w}_0$ stabilizes $\Delta$, and part c gives $v_0^{-1}{w}_0=1$. The longest element of a finite Coxeter group is therefore unique.

Let $R^{+}$ be the positive roots, $W$ the Weyl group, $\ell (w)$ its Coxeter length, $\rho$ the half-sum of positive roots, and ${\alpha }^{\vee }$ a coroot. For a dominant integral highest weight $\lambda$, the Weyl character formula isTaking the value at the identity gives the Weyl dimension formula

For the q-character convention relevant to the Principal sl2 subalgebra, set $h_{\mathrm{pr}}=2{\rho }^{\vee }$, so ${\alpha }_i(h_{\mathrm{pr}})=2$ for every simple root, and defineThe q-character formula is the principal specialization of the Weyl character formula:

Let $C$ be the fundamental chamber of a root system. The chambers $wC$ and $w{s}_{\alpha }C$ are adjacent across the reflecting hyperplane orthogonal to $w(\alpha )$. The chamber $wC$ lies on the side on which $w(\alpha )$ is positive. If $w(\alpha )\in \Pi$, then $C$ lies on that same side, so crossing this wall moves one step farther from $C$; if $w(\alpha )\in -\Pi$, it moves one step nearer. The gallery distance from $C$ to $wC$ is the Coxeter length $\ell (w)$, and adjacent chamber distances differ by one. Consequently

Write $N(w)$ for the inversion set of a Weyl-group element. Part b shows that $s_{\alpha }$ permutes $\Pi \setminus \{\alpha \}$. It follows that right multiplication by $s_{\alpha }$ changes the size of the inversion set byIndeed, all roots other than $\alpha$ are merely relabelled, while $w{s}_{\alpha }(\alpha )=-w(\alpha )$. Part c gives exactly the same recursion for the Coxeter length. Both quantities vanish at the identity, so induction along any word in the simple reflections gives

By the assumed transitivity on fundamental systems, some $w_0\in W$ sends $\Delta$ to $-\Delta$. It therefore sends the entire positive system of a root system $\Pi$ to $-\Pi$. Part d then givesFor every $w\in W$, its inversion set is contained in $\Pi$, so $\ell (w)\le |\Pi |$ and $w_0$ has maximal length.

If $u$ also has maximal length, then $N(u)=\Pi$, so $u(\Pi )=-\Pi$. Hence $w_0^{-1}u$ preserves $\Pi$ and has no inversions. Part d makes its Coxeter length zero, so it is the identity. Thus $u=w_0$, proving that the longest element of a finite Coxeter group is unique and has length $|\Phi |/2$.

Two linear extensions of a finite partially ordered set can be joined using as many admissible adjacent swaps as the inversion count of their relative permutation. Label the current list by the desired positions in the target. An adjacent descent consists of incomparable elements, since comparable elements have the same order in both lists. Swap that descent; its inversion count decreases by one. Iteration sorts the list and gives a path of minimal Coxeter length.

A reduced expression in a Coxeter group has the least possible Coxeter length among expressions in the generators of its Coxeter system.

A reduced expression for $w\in W$ is a product of the fewest possible simple generators representing $w$. The number of factors is its Coxeter length $\ell (w)$.