aa r X i v : . [ m a t h . C O ] N ov ON CORE AND BAR-CORE PARTITIONS
JEAN-BAPTISTE GRAMAIN AND RISHI NATH
Abstract. If s and t are relatively prime J. Olsson proved in [7] that the s -core of a t -core partition is again a t -core partition, and that the s -bar-core ofa t -bar-core partition is again a t -bar-core partition. Here generalized resultsare proved for partitions and bar-partitions when the restriction that s and t be relatively prime is removed. Introduction
The basic facts about partitions, hooks and blocks can be found in [3, Chapter 2]or [6, Chapter 1]. We recall a few key definitions here. A partition λ of n is definedas a non-increasing sequence of nonnegative integers ( λ , λ , · · · ) that sum to n. Apartition is represented graphically by its Young diagram [ λ ] , which consists of theset of nodes { ( i, j ) | ( i, j ) ∈ N , j ≤ λ i } . The node ( i, j ) is in the i th row and j thcolumn of [ λ ] . The rows of [ λ ] are labelled from top to bottom, while its columnsare labelled from left to right.To each node ( i, j ) in [ λ ] we associate the hook h ij of λ , which consists of thenode ( i, j ) itself, together with all the nodes { ( i, k ) | j < k } in [ λ ] (i.e. in the samerow as and to the right of ( i, j ) ), and all the nodes { ( ℓ, j ) | i < ℓ } (i.e. in thesame column as and below ( i, j ) ). The length of h ij is the total number of nodescontained in the hook. For any integer ℓ ≥ , we call ℓ -hook a hook of length ℓ , and ( ℓ ) -hook a hook of length divisible by ℓ . The information about the ( ℓ ) -hooks in λ is encoded in the ℓ -quotient q ℓ ( λ ) = ( λ , . . . , λ ℓ − ) of λ . The λ i ’s are partitionswhose sizes sum to the number w of ( ℓ ) -hooks in λ (called the ℓ -weight of λ ).The removal of an ℓ -hook h in λ is obtained by removing the ℓ nodes of [ λ ] in h , and migrating the disconnected nodes in [ λ ] up and to the left. The result is apartition of n − ℓ denoted by λ \ h . By removing all the ( ℓ ) -hooks in λ , one obtainsthe ℓ -core γ ℓ ( λ ) of λ . The partition γ ℓ ( λ ) contains no ( ℓ ) -hooks, and is uniquelydetermined by λ (i.e. doesn’t depend on the order in which we remove the ℓ -hooksin λ ). The partition λ is entirely determined by its ℓ -core and ℓ -quotient.It is well-known that the irreducible complex characters of the symmetric group S n are labelled by the partitions of n . If p is a prime, then the distribution ofirreducible characters of S n into p -blocks has a combinatorial description knownas the Nakayama Conjecture: two characters χ λ , χ µ ∈ Irr( S n ) belong to the same p -block if and only if λ and µ have the same p -core (see [3, Theorem 6.1.21]). Hencewe define, for each integer ℓ ≥ , an ℓ -block of partitions of n to be the set of allpartitions of n having a common given ℓ -core. We now recall the analogous notions and results for bar-partitions, which canbe found in [6, Chapter 1]. A bar-partition is a partition λ comprised of distinctparts. To each bar-partition we associate a shifted Young diagram S ( λ ) obtainedby shifting the i th row of the usual Young diagram ( i − positions to the right.The j -th node in the i -th row will be called the ( i, j ) -node. To each node ( i, j ) in S ( λ ) , one can associate a bar and bar-length . For any odd integer ℓ , a bar-partition λ is entirely determined by its ¯ ℓ -core ¯ γ ℓ ( λ ) and its ¯ ℓ -quotient ¯ q ℓ ( λ ) . The bar-core ¯ γ ℓ ( λ ) is obtained by removing from λ all the bars of length divisible by ℓ (called ( ℓ ) -bars). The bar-quotient of λ is of the form ¯ q ℓ ( λ ) = ( λ , λ , . . . λ ( ℓ − / ) , where λ is a bar-partition, λ , ..., λ ( ℓ − / are partitions, and the sizes of the λ i ’s sumto the number of ( ℓ ) -bars in λ (called ¯ ℓ -weight of λ ).It is well-known that the bar-partitions of n label the faithful irreducible complexcharacters of the 2-fold covering group ˜ S n of S n . These correspond to irreducibleprojective representations of S n , and are known as spin-characters . If p is anodd prime, then the distribution of spin-characters of ˜ S n of positive defect into p -blocks has a combinatorial description known as the Morris Conjecture: twospin-characters of ˜ S n of positive defect belong to the same p -block if and only ifthe bar-partitions labelling them have the same ¯ p -core (see [6, Theorem 13.1]).In analogy with this, we define, for each odd integer ℓ ≥ , an ¯ ℓ -block of partitions of n to be the set of all bar-partitions of n having a common given ¯ ℓ -core.2. Some new results on cores and bar-cores
In this section, we generalize to arbitrary integers s and t the results on cores andbar-cores proved by J. B. Olsson in [7] when s and t are coprime. Note that Olsson’sresult ([7, Theorem 1]) was interpreted by M. Fayers through alcove geometry andactions of the affine symmetric group (see [2]). It was also used by F. Garvan andA. Berkovich to bound the number of distinct values their partition statistic (theGBG-rank) can take on a t -core (mod s ) (see [1, Theorem 1.2]).We keep the notation as in Section 1. Theorem 2.1.
For any two positive integers s and t , the s -core of a t -core partitionis again a t -core partition. Remark 2.2.
This result was proved by J. B. Olsson in [7] , under the extra hy-pothesis that s and t are relatively prime. R. Nath then gave in [5] a proof of theresult in general. We give here another proof which, unlike the one given by Nath,uses Olsson’s result, and provides the framework for the proof for bar-partitions.Proof. Consider a t -core partition λ . Let g = gcd( s, t ) , and write s = s/g and t = t/g . It’s a well-known fact (see e.g. [6, Theorem 3.3]) that there is a canonicalbijection ϕ between the set of hooks of length divisible by g in λ and the set ofhooks in q g ( λ ) = ( λ , . . . , λ g − ) (i.e. hooks in each of the λ i ’s). For each positiveinteger k and hook h of length kg in λ , the hook ϕ ( h ) has length k . Furthermore,we have q g ( λ \ h ) = q g ( λ ) \ ϕ ( h ) .In particular, since λ is an t -core, and since t = t g , we see that q g ( λ ) containsno t -hook, so that each λ i is an t -core.Now, the s -hooks in λ are in bijection with the s -hooks in q g ( λ ) . When weremove them all, we obtain that the s -core γ s ( λ ) has g -core γ g ( γ s ( λ )) = γ g ( λ ) and g -quotient q g ( γ s ( λ )) = ( γ s ( λ ) , . . . γ s ( λ g − )) . But, since s and t are coprime,the s -core of each t -core λ i is again a t -core ([7, Theorem 1]). This shows that ENERALIZED RESULTS ON CORE PARTITIONS 3 q g ( γ s ( λ )) has no t -hook, which in turn implies that γ s ( λ ) contains no t -hook,whence is an t -core. (cid:3) As we mentionned in Section 1, when p is a prime, the study of p -cores is linkedto that of the p -modular representation theory of the symmetric group S n (asthey label the p -blocks of irreducible characters). When ℓ ≥ is an arbitraryinteger, it turns out that it is still possible to describe an ℓ -modular representationtheory of S n (see [4]). The theory of ℓ -blocks obtained in this way is in factrelated to the ordinary representation theory of an Iwahori-Hecke algebra of type S n , when specialized at an ℓ -root of unity. Külshammer, Olsson and Robinsonproved in [4] the following analogue of the Nakayama Conjecture: two characters χ λ , χ µ ∈ Irr( S n ) belong to the same ℓ -block if and only if λ and µ have the same ℓ -core.It is therefore legimitate to study ℓ -cores and ℓ -blocks of partitions. In particular,we obtain from Theorem 2.1 a generalization of [7, Corollary 3]. We call principal ℓ -block of n the ℓ -block of partitions of n which contains the partition ( n ) (i.e. theset of partitions labelling the characters of the principal ℓ -block of S n ). Corollary 2.3.
Let r , s and t be any positive integers such that s > r ≥ t , and let n = as + r for some a ∈ Z ≥ . Then the principal s -block of n contains no t -core.Proof. Suppose the partition λ of n is a t -core. The s -core γ of λ , which is obtainedby removing s -hooks, must therefore be a partition of some m which differs from n by a multiple of s , i.e. m = bs + r for some b such that a ≥ b ≥ . By Theorem2.1, γ is also a t -core. Now, if λ was in the principal s -block of n , then its s -corewould be the same as that of the cycle ( n ) , hence also a cycle. We would thus have γ = ( m ) . But since m ≥ r ≥ t , the cycle ( m ) contains a t -hook, hence cannot be a t -core. (cid:3) In terms of blocks of characters, this means that, if s , t and n are as above, thenthere is no trivial block inclusion of a t -block in the principal s -block of S n (see[8]).We now prove the analogue results for bar-cores, which was proved by Olssonwhen s and t are odd and coprime ([7, Theorem 4]). Theorem 2.4.
For any two odd positive integers s and t , the ¯ s -core of an ¯ t -corepartition is again a ¯ t -core partition.Proof. Take any ¯ t -core λ . Let g = gcd( s, t ) , and write s = s/g and t = t/g .There is a canonical bijection ϕ between the set of bars of length divisible by g in λ and the set of bars in its ¯ g -quotient ¯ q g ( λ ) = ( λ , , λ , . . . , λ ( g − / ) , where a barin ¯ q g ( λ ) is either a bar in the bar-partition λ or a hook in one of the partitions λ , . . . , λ ( g − / (see [6, Theorem 4.3]). For each positive integer k and bar b oflength kg in λ , the bar ϕ ( b ) has length k . Furthermore, we have ¯ q g ( λ \ b ) =¯ q g ( λ ) \ ϕ ( b ) .The same argument as in the proof of Theorem 2.1 thus proves that λ is an ¯ t -core, that each λ i ( ≤ i ≤ ( g − / ) is an t -core, and that the ¯ s -core ¯ γ s ( λ ) of λ has ¯ g -quotient ¯ q g (¯ γ s ( λ )) = (¯ γ s ( λ ) , γ s ( λ ) , . . . γ s ( λ ( g − / )) . And, since s and t are coprime, the s -core of each t -core λ i ( ≤ i ≤ ( g − / ) is again a t -core ([7, Theorem 1]), and the ¯ s -core of the ¯ t -core λ is again a ¯ t -core ([7, JEAN-BAPTISTE GRAMAIN AND RISHI NATH
Theorem 4]). This shows that the ¯ g -quotient of ¯ γ s ( λ ) contains no t -bar, whichfinally implies that ¯ γ s ( λ ) contains no t -bar, whence is an ¯ t -core. (cid:3) In analogy with the partition case, we call principal ¯ ℓ -block of bar-partitions of n (for ℓ odd) the ¯ ℓ -block containing the bar-partition ( n ) . Then the same argumentas for the proof of Corollary 2.3 yields Corollary 2.5.
Let r , s and t be any positive integers such that s and t are oddand s > r ≥ t , and let n = as + r for some a ∈ Z ≥ . Then the principal ¯ s -block of n contains no ¯ t -core. References [1] A. Berkovich and F. Garvan. The GBG-rank and t -cores I. Counting and 4-cores. J. Comb.Number Theory , 1(3):237–252, 2009.[2] M. Fayers. The t -core of an s -core. Preprint .[3] G. James and A. Kerber.
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