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On graded Cartan invariants of symmetric groups and Hecke algebras

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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

  1. 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)

\(I^v_\ell (\lambda ), J^v_\ell (\lambda )\) :

Laurent polynomials in Definition 1.8 (1.4)

\({{\mathrm{\mathsf {max-Spec}}}}(R)\) :

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)

\(a_{\theta }^{(p)}(n)\) :

A rational number from Definition 2.8 (2.2)

\({{\mathrm{\mathsf {Sym}}}}^m\) :

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 )

\(S^d (A)\) :

Matrix in Definition 2.13 (2.3)

\(\Lambda _\ell =\bigotimes _{t=1}^\ell \Lambda ^{(t)}\) :

\(\ell \)-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)

\(M_{\ell ,d}, K_{\ell ,d}\) :

Transition matrices in Definition 2.14 (2.3)

\({\mathcal {P}}, {\mathcal {P}^\vee }\) :

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)

\(\langle \cdot ,\cdot \rangle _{\mathsf {QSh}}, \langle \cdot ,\cdot \rangle _{\mathsf {RSh}}\) :

Versions of Shapovalov form from Proposition 3.2 (3.1)

\(P(\lambda )\) :

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)

\(\mathsf {QSh}^\mathsf {M}_{\lambda ,\mu }, \mathsf {RSh}^\mathsf {M}_{\lambda ,\mu }\) :

Gram matrices of Shapovalov forms, see Definition 3.3 (3.1)

\(\fallingdotseq \) :

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)

\(\equiv ^F_R\) :

Fitting equivalence of matrices, see Definition 4.1 (4.1)

\({{\mathrm{\mathsf {Fitt}}}}_d (T)\) :

d-th Fitting ideal of a matrix T (4.1)

\(\Phi _n, \Psi _n\) :

Cyclotomic polynomial and its scaled version (4.2)

\(\rho _z^{(p)}\) :

Function from Definition 4.5 (4.2)

\(\varphi _{s,n}\) :

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)

\(\lambda ^{<r},\lambda ^{\ge r},\overline{\lambda }^{\, r}\) :

p-power partitions from Definition 5.6 (5.2)

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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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Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    Anton Evseev

  2. Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Meguro, Tokyo, 153-8914, Japan

    Shunsuke Tsuchioka

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  1. Anton Evseev
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  2. Shunsuke Tsuchioka
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Correspondence to Shunsuke Tsuchioka.

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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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