Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 111 (Coxeter Groups)

A fundamental system of a root system is a subset $\Delta \subset \Phi$ which is a basis of $V$ and for which every $\beta \in \Phi$ has an expansionwhose coefficients are either all nonnegative or all nonpositive. Its associated positive system of a root system isThus $\Phi =\Pi \bigsqcup (-\Pi )$, and the elements of $\Delta$ are the simple roots.

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The Weyl reflection in $\alpha$ satisfiesNow take $\beta \in \Pi \setminus \{\alpha \}$ and expand it in the basis $\Delta$. At least one coefficient belonging to a simple root other than $\alpha$ is positive. Sincethe reflection changes only the coefficient of $\alpha$. Every root has simple-root coefficients of one sign, so the unchanged positive coefficient prevents $s_{\alpha }(\beta )$ from being negative. Hence $s_{\alpha }(\beta )\in \Pi$. Because $s_{\alpha }$ is an involution, it permutes $\Pi \setminus \{\alpha \}$, while it exchanges $\alpha$ and $-\alpha$. Therefore

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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

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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

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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 $|\Phi |/2$.

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Let $V$ be the real vector space with basis $(e_i{)}_{i\in I}$. The Coxeter Gram matrix defines the symmetric bilinear formIts diagonal entries are $2$. The Geometric representation of a Coxeter group is generated by the reflectionsEach has square one, and on $\mathrm{span}\{e_i,e_j\}$ the product $\sigma (x_i)\sigma (x_j)$ has order $m_{ij}$. The reflections therefore satisfy the Coxeter relations and define a group representation $\sigma :W\to GL(V)$.

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The Coxeter graph has vertex set $I$. Distinct vertices $i,j$ are joined precisely when $m_{ij}\ge 3$, and the edge is labelled $m_{ij}$ when $m_{ij}>3$; the customary unlabelled edge therefore means $m_{ij}=3$. The Coxeter system is an Irreducible Coxeter system precisely when this graph is connected.

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PutAfter ordering $I_1$ before $I_2$, the only nonzero off-diagonal entries between the two blocks occur at $(i,j)$ and $(j,i)$, where they equal $-a$. Thuswhere the coordinate vectors in the two blocks are understood. Expanding the determinant according to whether neither or both cross-block entries are selected givesThe minus sign is the sign of the transposition pairing the two cross-block entries. This formula remains valid when either diagonal block is singular, so no inverse or Schur complement is needed.

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All edges in the Type E(p,q,k) Coxeter graph are unlabelled, so the factor $a^2$ in part c is one. Separate the arm of length $p$ from the central vertex. The remaining two arms form a type $A_{q+k+1}$ chain, while deleting the central vertex leaves the disjoint type $A_q$ and type $A_k$ chains. Using $\det G(A_r)=r+1$ in the formula from part c yieldsThe associated bilinear form is degenerate exactly whenThe positive-integer solutions of $1/a+1/b+1/c=1$, up to permutation, are $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$. Consequently the degenerate arm-length triples are the permutations ofFor every other allowed $(p,q,k)$ the determinant is nonzero, so the form is nondegenerate.

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The Generic Hecke algebra of a Coxeter system is the free module with basis $(T_w{)}_{w\in W}$ over the polynomial ring in parameters $a_i$, subject to $a_i=a_j$ whenever $x_i$ and $x_j$ are conjugate, and with multiplicationEquivalently, its generators satisfy the Coxeter braid relations andIn type $A_n$, all simple generators are conjugate, so there is one parameter.

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A BN-pair consists of subgroups $B,N\le G$ for which $G=⟨B,N⟩$, the subgroup $H=B\cap N$ is normal in $N$, the quotient $W=N/H$ is generated by a distinguished set $S$ of involutions, and the Bruhat multiplication and nondegeneracy axioms hold. The quotient is the associated Weyl group, and the axioms give the Bruhat decomposition of a BN-pair

The Iwahori-Hecke algebra of a BN-pair may be defined, up to the usual opposite-algebra convention, byIts standard basis $(T_w{)}_{w\in W}$ is indexed by the Bruhat double cosets. For a simple generator $x_i$ represented by ${\dot{x}}_i\in N$, setThe double-coset multiplication rule is the generic rule from part a with $a_i$ specialized to $q_i\cdot 1_k$. Thus $H_k(G,B)$ is a specialization of the generic algebra.

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Every Hecke parameter of a BN-pair $q_i$ divides $|B|$ and, because $B$ is a $q$-group, is a power of $q$. If $q\equiv 1(\mathrm{mod}p)$, then $q_i=1$ in the field $k$ of characteristic $p$. The specialized quadratic relation becomeswhile the braid relations are unchanged. These are the defining relations of the Coxeter group $W$, so $x_i\mapsto T_i$ induces a surjective homomorphismBoth algebras have bases indexed by $W$, hence the homomorphism is an isomorphism of group algebras.

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There is a missing hypothesis in the printed claim: it is false when every irreducible component of $W$ has type $A_1$. The intended statement holds as soon as $W$ has an irreducible component of rank at least two, which we now assume.

Since $q=p$, every Hecke parameter of a BN-pair vanishes in $k$, and $H$ is the 0-Hecke algebra withLet $w_0$ be the Longest element of a finite Coxeter group. Choose a simple generator $s$ in a component of rank at least two, putThe element $r$ is again a simple generator. The identities $rv=w_0$ and $\ell (w_0)=\ell (v)+1$ giveIf $t\ne r$ is simple, then $t$ is a left descent of both $v$ and $w_0$: using $\ell (w_{0}u)=\ell (w_0)-\ell (u)$, one gets $\ell (tv)=\ell (v)-1$. HenceThe one-dimensional subspace $kX$ is therefore a left ideal. It is nonzero because $T_v$ and $T_{w_0}$ are distinct basis elements.

Every reduced expression in a Coxeter group for $v=w_{0}s$ contains $r$: in an irreducible finite component of rank at least two, deleting one final generator from $w_0$ does not remove any vertex from its support. A reduced expression for $w_0$ contains $r$ as well. Since $T_{r}X=0$, associativity now givesIf $H$ were a semisimple algebra, the left ideal $kX$ would be a direct summand of the regular module. The corresponding projection would produce a nonzero idempotent in $kX$, impossible because $(kX{)}^2=0$. Thus $H$ is not semisimple.

For completeness, if $W\cong (A_1{)}^m$, thenwhich is semisimple. This is the counterexample showing why the omitted rank condition is necessary.

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The Coxeter number is the order of a Coxeter element. In the stated families the values areHere $D_{2n}$ is the dihedral group of order $2n$, $S_n$ is the symmetric group, $S{S}_n$ is the signed symmetric group, and $ES{S}_n$ is the even signed symmetric group.

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SetIn the Geometric representation of a Coxeter group,including $i=j$, since $a_{ii}=2\cos \pi =-2$. As ${\beta }_i=\sigma (c_{i-1})e_i$, telescoping givesSumming from $i=1$ to $j-1$ yieldsor equivalentlySumming the same telescoping identity all the way to $n$ and substituting this first formula giveswhich is the second required identity.

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Let $B$ be the matrix whose $i$th column consists of the coordinates of ${\beta }_i$ in the basis $(e_j)$. Define the upper-triangular matrix $U$ and lower-triangular matrix $L$ byThe first identity in part b says $I=BU$, so $B=U^{-1}$. The second says that the matrix $C$ of $\sigma (c)$ is $C=BL=U^{-1}L$. Since $U$ has diagonal entries one, $\det U=1$, and thereforeThus $\det (tU-L)$ is the characteristic polynomial of the Coxeter element in its geometric representation.

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At $t=1$, the diagonal entries of $U-L$ are $2$, while every off-diagonal $(i,j)$ entry isHence $U-L$ is exactly the Coxeter Gram matrix $G(W)$, and part c givesFor a Finite Coxeter group the Gram matrix is positive definite, and for a Hyperbolic Coxeter group it is nondegenerate with Lorentzian signature. In either case $\det (I-C)\ne 0$, so $1$ is not an eigenvalue of $C$ and the Coxeter element fixes no nonzero vector.

For an Affine Coxeter group, the Gram form has a nonzero radical. If $0\ne v\in \mathrm{rad}G(W)$, then $⟨v,e_i⟩=0$ for every $i$, and every generating reflection satisfiesTheir product $\sigma (c)$ therefore fixes $v$. Thus every affine Coxeter element has a nonzero fixed vector.

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