Abstract
We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A at roots of unity. These matrices are \({\mathbb {Z}}[v,v^{-1}]\)-valued and may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring \({\mathbb {Z}}[v,v^{-1}]\). This proves and generalizes a conjecture of Ando-Suzuki-Yamada on the invariants of these matrices over \({\mathbb {Q}}[v,v^{-1}]\) and also generalizes the first author’s recent proof of the Külshammer-Olsson-Robinson conjecture over \({\mathbb {Z}}\).
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Notes
Our \(x_r^{(i)}\) and \(y_r^{(i)}\) correspond respectively to \(\tilde{P}^-_{i,r}\) and \(h'_{i,-r}\) in loc. cit.
Abbreviations
- \({\mathfrak {S}}_n\) :
-
Symmetric group (1.1)
- \({\mathcal {H}}_n ({\mathbb {F}};q)\) :
-
Hecke algebra (1.1)
- \(\eta _{\ell } \in k_{\ell }\) :
-
A primitive \(\ell \)-th root of unity in a field (1.1)
- \({\mathsf {Mod}({A})}\) :
-
The category of finite-dimensional left A-modules (1.1)
- \({{\mathrm{\mathsf {PC}}}}(D)\) :
-
Projective cover of D (1.1)
- \(C_A\) :
-
Cartan matrix of an algebra A (1.1)
- \(\equiv _R\) :
-
Unimodular equivalence of matrices (1.2)
- \(\ell _k\) :
-
\(\ell /(\ell ,k)\) (1.2)
- \(I^v_\ell (\lambda ), J^v_\ell (\lambda )\) :
- \({{\mathrm{\mathsf {max-Spec}}}}(R)\) :
-
The set of maximal ideals of a ring R (1.7.1)
- \({{\mathrm{\mathsf {Mat}}}}_\ell (R), {{\mathrm{\mathsf {Mat}}}}_S(R)\) :
-
Matrix algebra (1.7.2)
- \(1_S\) :
-
Identity matrix (1.7.2)
- \({{\mathrm{\mathsf {diag}}}}(\{r_s \mid s\in S\})\) :
-
Diagonal matrix (1.7.2)
- \(\bigoplus _i M_i\) :
-
Block-diagonal matrix (1.7.2)
- \(\nu _p\) :
-
p-adic valuation (1.7.3)
- \({\mathbb {N}}\) :
-
The set of nonnegative integers (1.7.4)
- \(\mathsf {Prm}\) :
-
The set of prime numbers (1.7.4)
- \(n_{\Pi }\) :
-
\(\Pi \)-part of n (1.7.4)
- \(\Pi ', p'\) :
-
The complements of \(\Pi , \{p\}\) in \(\mathsf {Prm}\) (1.7.4)
- (a, b):
-
Greatest common divisor of a and b (1.7.4)
- \({\mathbf {k}}\) :
-
The function field \({\mathbb {Q}}(v)\) (1.7.5)
- \(\mathscr {A}\) :
-
The ring of Laurent polynomials \({\mathbb {Z}}[v,v^{-1}]\) (1.7.5)
- \(\mathsf {bar}\) :
-
Bar involution \(v\mapsto v^{-1}\) on \({\mathbf {k}}\) (1.7.5)
- \({{\mathrm{\mathsf {Infl}}}}_t\) :
-
The inflation map \(v\mapsto v^t\) on \(\mathscr {A}\) (1.7.5)
- \({[n]}_m\) :
-
Quantum integer (1.7.5)
- \({[n]}_m !\) :
-
Quantum factorial (1.7.5)
- \(\equiv _G\) :
-
Conjugacy relation in a group G (1.7.6)
- \(g_p, g_{p'}\) :
-
p-part and \(p'\)-part of g (1.7.6)
- \(m_k (\lambda )\) :
-
Multiplicity of k in a partition \(\lambda \) (1.7.7)
- \(\ell (\lambda )\) :
-
Length of a partition \(\lambda \) (1.7.7)
- \(|\lambda |\) :
-
Size of a partition \(\lambda \) (1.7.7)
- \(\mathsf {Par}, \mathsf {Par}(n)\) :
-
Set of partitions (1.7.7)
- \(\mathsf {CRP}_s(n)\) :
-
The set of s-class regular partitions of n (1.7.7)
- \(\mathsf {RP}_s (n)\) :
-
The set of s-regular partitions of n (1.7.7)
- \(\mathsf {Par}_m (n)\) :
-
The set of m-multipartitions of n (1.7.7)
- \(\mathsf {Par}_p (n,\nu )\) :
-
The set of partitions of n with “\(p'\)-part” \(\nu \) (1.7.7)
- \(\mathsf {Pow}_p (n)\) :
-
The set of p-power partitions of n (1.7.7)
- \(\lambda +\mu \) :
-
Sum of two partitions (1.7.7)
- \(\Lambda \) :
-
Ring of symmetric functions (2.1)
- \(\chi _V\) :
-
Character of \({\mathfrak {S}}_n\) afforded by module V (2.1)
- \(p_\mu , p_k\) :
-
Power sum symmetric functions (2.1)
- \(C_\mu \) :
-
Conjugacy class corresponding to a partition \(\mu \) (2.1)
- \(z_{\mu }\) :
-
Order of centralizer of an element of \(C_\mu \) (2.1)
- \(\mathsf {ch}\) :
-
Isometry between a Grothendieck group and symmetric functions (2.1)
- \({\mathfrak {S}}_\lambda \) :
-
Parabolic subgroup of \({\mathfrak {S}}_n\) (2.1)
- \(\mathsf {triv}_{{\mathfrak {S}}_\lambda }\) :
-
Trivial representation of \({\mathfrak {S}}_\lambda \) (2.1)
- \(M_n\) :
-
Table of permutation characters of \({\mathfrak {S}}_n\) (2.1)
- \(h_{\mu }\) :
-
Complete symmetric function (2.1)
- \(m_\mu \) :
-
Monomial symmetric function (2.1)
- \({\mathcal {M}}_{\lambda ,\mu }\) :
-
A certain set of size \(M_{\lambda ,\mu }\) (2.1)
- \(N_n^{(p)}\) :
-
“p-local” submatrix of \(M_n\) (2.2)
- \(L_n^{(p)}\) :
-
A certain block-diagonal matrix (2.2)
- \(a_{\theta }^{(p)}(n)\) :
- \({{\mathrm{\mathsf {Sym}}}}^m\) :
-
Symmetric power functor (2.3)
- \({{\mathrm{\mathsf {Mult}}}}_m(\ell )\) :
-
The set of weakly increasing m-tuples of elements of \(\{1,\dots ,\ell \}\) (2.3)
- \(\Omega _{\ell ,d}\) :
-
A set of tuples, which is in bijection with \(\mathsf {Par}_{\ell } (d)\) (2.3 )
- \(S^d (A)\) :
- \(\Lambda _\ell =\bigotimes _{t=1}^\ell \Lambda ^{(t)}\) :
-
\(\ell \)-colored ring of symmetric functions (2.3)
- \(m_{\mu }^{(t)}, h_\mu ^{(t)}, p_{\mu }^{(t)}\) :
-
Images of \(m_\mu , h_\mu , p_\mu \) in \(\Lambda ^{(t)}\) (2.3)
- \(M_{\ell ,d}, K_{\ell ,d}\) :
- \({\mathcal {P}}, {\mathcal {P}^\vee }\) :
-
Weight lattice and its dual (3.1)
- \(\Pi , \Pi ^\vee \) :
-
Sets of simple roots and corresponding coroots (3.1)
- \(Q^+\) :
-
Positive part of the root lattice (3.1)
- \({\mathcal {P}}^+\) :
-
Set of dominant integral weights (3.1)
- \(\Lambda _i\) :
-
A dominant integral weight (3.1)
- \(W=W(X)\) :
-
Weyl group (3.1)
- \(U_v = U_v (X)\) :
-
Quantum group (3.1)
- \(U_v^+, U_v^0, U_v^-\) :
-
Subalgebras in the triangular decomposition of \(U_v\) (3.1)
- \(V(\lambda )\) :
-
Highest weight module (3.1)
- \(1_{\lambda }\) :
-
Highest weight vector (3.1)
- \(\langle \cdot ,\cdot \rangle _{\mathsf {QSh}}, \langle \cdot ,\cdot \rangle _{\mathsf {RSh}}\) :
- \(P(\lambda )\) :
-
The set of weights of \(V(\lambda )\) (3.1)
- \(V(\lambda )_\mu \) :
-
\(\mu \)-weight space of \(V(\lambda )\) (3.1)
- \((U_v^-)^\mathscr {A}\) :
-
An \(\mathscr {A}\)-lattice in \(U_v^-\) (3.1)
- \(V(\lambda )^{\mathscr {A}}, V(\lambda )_\nu ^\mathscr {A}\) :
-
\(\mathscr {A}\)-lattices in \(V(\lambda ), V(\lambda )_\nu \) (3.1)
- \(\mathsf {QSh}^\mathsf {M}_{\lambda ,\mu }, \mathsf {RSh}^\mathsf {M}_{\lambda ,\mu }\) :
- \(\fallingdotseq \) :
-
An equivalence relation on matrices (3.1)
- \(\widehat{X}\) :
-
Extended Cartan matrix of X (3.2)
- \(\delta \) :
-
Null root (3.2)
- \({\mathsf {Mod}_{\mathsf {gr}}({A})}\) :
-
Category of finite-dimensional graded A-modules (3.3)
- \(M_n\) :
-
Graded n-component of a graded module M (3.3)
- \(M\langle k\rangle \) :
-
Graded module M with grading shifted down by k (3.3)
- \(\mathcal S(A)\) :
-
Set of representatives of simple graded A-modules (3.3)
- \(C_A^v\) :
-
Graded Cartan matrix of A (3.3)
- \(\mathsf {Proj}_\mathsf {gr}({A})\) :
-
Category of projective graded A-modules (3.3)
- \(\mathsf {Bl}_\ell (n)\) :
-
A certain set of pairs \((\rho ,d)\) where \(\rho \) is an \(\ell \)-core (3.3)
- \(\mathsf {Cok}_{T}\) :
-
Cokernel of the map given by a matrix T (4.1)
- \(\equiv '_R\) :
-
Unimodular pseudo-equivalence of matrices, see Definition 4.1 (4.1)
- \(\equiv ^F_R\) :
- \({{\mathrm{\mathsf {Fitt}}}}_d (T)\) :
-
d-th Fitting ideal of a matrix T (4.1)
- \(\Phi _n, \Psi _n\) :
-
Cyclotomic polynomial and its scaled version (4.2)
- \(\rho _z^{(p)}\) :
- \(\varphi _{s,n}\) :
-
A bijection from s-regular to s-class regular partitions (4.2)
- \(\beta _M\) :
-
Auto-bijection of \(\mathsf {Par}\) from Definition 4.6 (4.2)
- \(g_{k,t}^{(\ell ,p)}, f_{k,t}^{(\ell )}, I_{\ell ,p}^v (\lambda )\) :
-
Certain products of quantum integers, see Definition 4.7 (4.2)
- \({\mathcal {F}}_{k,t,z}^{(\ell ,p)}, {\mathcal {G}}_{k,t,z}^{(\ell ,p)}\) :
-
Certain sets of integers related to \(f_{k,t}^{(\ell )}, g_{k,t}^{(\ell ,p)}\) (4.2)
- \(\lambda ^{<r},\lambda ^{\ge r},\overline{\lambda }^{\, r}\) :
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Acknowledgments
S.T. thanks Yuichiro Hoshi, Yoichi Mieda and Hiraku Kawanoue for discussions on §6. In particular, Theorem 6.5 is due to Kawanoue (see Remark 6.8).
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The list of symbols gives references to subsections where symbols are defined.
Anton Evseev was partially supported by the EPSRC Postdoctoral Fellowship EP/G050244 and the EPSRC grant EP/L027283. Shunsuke Tsuchioka was supported in part by JSPS Kakenhi Grants 11J08363 and 26800005.
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Evseev, A., Tsuchioka, S. On graded Cartan invariants of symmetric groups and Hecke algebras. Math. Z. 285, 177–213 (2017). https://doi.org/10.1007/s00209-016-1703-0
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DOI: https://doi.org/10.1007/s00209-016-1703-0
Keywords
- Symmetric groups
- Hecke algebras
- Graded representation theory
- Modular representation theory
- Külshammer-Olsson-Robinson conjecture
- Khovanov-Lauda-Rouquier algebras
- Generalized blocks
- Categorification
- Lie theory
- Quantum groups