Super Major Index Cyclic Sieving

SUPER MAJOR INDEX CYCLIC SIEVING STEPHAN PF ANNERER Abstract. Recen tly , Armon and Sw anson in troduced signed standard tableaux and a corre- sponding super ma jor index that refines the classical ma jor index. In this paper, we prov e that signed standard tableaux of rectangular shape exhibit a cyclic sieving phenomenon (CSP) under the com bined action of Sc h¨ utzenberger promotion and cyclic shift of the signs, with the sieving polynomial given by the sup er ma jor index generating function. This extends Rhoades’s celebrated CSP for standard Y oung tableaux. F urthermore, b y considering Carte- sian products of tableaux, we generalize this result to arbitrary non-rectangular shapes. 1. Introduction The cyclic sieving phenomenon (CSP), introduced by Reiner, Stan ton, and White [RSW04], rev eals a surprising connection b et ween the enumeration of symmetry classes and the ev aluation of q -analogues at roots of unity . This phenomenon generalizes Stem bridge’s earlier “ q = − 1 phenomenon” [Ste94] and has since b een found in diverse combinatorial contexts (see [Sag11] for a surv ey). A celebrated instance of the CSP in volv es standar d Y oung table aux (SYT). F or a rectan- gular partition λ , Rhoades [Rho10] prov ed that the set SYT( λ ) exhibits a CSP under the action of Sch¨ utzen b erger’s pr omotion , with the sieving p olynomial given b y the q -analogue of the F rame–Robinson–Thrall hook length formula [FR T54], which agrees up to a simple twist with the major index generating series. While promotion is well-behav ed on rectangles, its order and structure on arbitrary shap es are more complex. In recent join t work with Alexandersson et al. [APRU21] we established an “implicit” CSP for general shap es with the same sieving p olynomial, though the explicit bijective action remains conjectural or completely unkno wn in man y cases. In this pap er, we extend the study of cyclic sieving to signe d standar d table aux 1 , ob jects recen tly inv estigated by Armon and Sw anson [AS25] in the context of Lie superalgebras gl ( m | n ). A signed standard tableau extends the classical definition by distinguishing a subset of “negative” en tries. They introduced a sup er major index statistic on these tableaux, which refines the classical ma jor index and encodes the in terplay b et ween descen ts and negativ e entries. Our main result establishes that rectangular signed standard tableaux together with the sup er ma jor index generating series exhibit the cyclic sieving phenomenon under a super-analogue of promotion. Theorem 1.1 (Main Theorem A) . L et λ ⊢ n b e a r e ctangular p artition. F or any 0 ≤ k ≤ n , the triple  SYT ± k ( λ ) , ⟨ pr ⟩ , q γ ( n,k ) − κ ( λ ) f λ ± k ( q )  exhibits the cyclic sieving phenomenon, wher e the action is Sch ¨ utzenb er ger pr omotion on the table au and cyclic shift on the ne gative set. The sieving p olynomial f λ ± k ( q ) is the gener ating function for the sup er major index over signe d table aux with k ne gative entries, and the twist κ ( λ ) is an inte ger dep ending on the shap e of λ and γ ( n, k ) = ( n − 1)  k 2  . W e also consider the case of arbitrary shap es. While a CSP may not exist for a single cop y of SYT( λ ) due to negativ e character ev aluations, [APRU21] hav e sho wn that for any shap e λ , a CSP exists for the m -fold Cartesian product SYT( λ ) m pro vided the polynomial ev aluations are non-negativ e (whic h is alwa ys true if m is ev en). W e also extend this result to the signed case. Date : March 18, 2026. 1 Armon and Swanson call them standard sup er table aux , but that term is ambiguous in literature 1 2 STEPHAN PF ANNERER Theorem 1.2 (Main Theorem B) . L et λ b e a p artition of n . F or any m ≥ 1 , ther e exists a cyclic gr oup action of or der n on SYT ± k ( λ ) m such that the triple (SYT ± k ( λ ) m , C n , ( q γ ( n,k ) f λ ± k ( q )) m ) exhibits the cyclic sieving phenomenon if and only if for al l d | n , the evaluation ( f λ ( ξ d )) m is a non-ne gative inte ger, wher e ξ is a primitive n -th r o ot of unity. In p articular, if m is even, such a CSP always exists. The pap er is organized as follows. In Section 2, we review the necessary background on stan- dard Y oung tableaux, the cyclic sieving phenomenon, and the connection b et ween ev aluations of ma jor index polynomials at ro ots of unity and the en umeration of b order strip tableaux. In Section 3, we introduce signed standard tableaux and the sup er ma jor index, and w e presen t a k ey factorization of its generating function. Section 4 contains the pro of of our main results; we state and pro ve a unified theorem and sho w ho w b oth main theorems follow as special cases. 2. Back ground 2.1. Standard Y oung tableaux. A p artition of a p ositiv e integer n is a s equence λ = ( λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ > 0) of p ositiv e in tegers whose sum is n . The Y oung diagram of shap e λ is a left-justified arra y of λ i b o xes in row i . If the Y oung diagram of λ is a rectangle, i.e., λ = ( b, b, . . . , b | {z } a copies ) for some a, b ≥ 1, w e sa y λ is a r e ctangular p artition and write λ = a × b . If λ and µ are partitions, we say λ c ontains µ if the Y oung diagram of µ is contained in that of λ . In this case, the skew shap e λ/µ is the set-theoretic difference of the t wo diagrams. A standar d Y oung table au (SYT) of shap e λ is a filling of the b o xes of the Y oung diagram with the n umbers 1 , 2 , . . . , n , eac h used exactly once, such that the entries strictly increase left- to-righ t in each ro w and strictly increase top-to-b ottom in eac h column. W e denote the set of standard Y oung tableaux of shap e λ b y SYT( λ ). Giv en a tableau T ∈ SYT( λ ), an integer i ∈ { 1 , 2 , . . . , n − 1 } is called a desc ent of T if the en try i + 1 appears in a strictly lo wer ro w than the en try i . The desc ent set of T is DES( T ) = { i ∈ { 1 , . . . , n − 1 } : i + 1 lies in a lo wer row than i } . The major index of T is the sum of its descents: ma j( T ) = X i ∈ DES( T ) i. W e denote the generating function for the ma jor index by f λ ( q ) : = X T ∈ SYT( λ ) q ma j( T ) . Example 2.1. C onsider the following standard Y oung tableau: T = 1 2 4 3 5 6 7 The descent set is DES( T ) = { 2 , 4 , 5 } . The ma jor index is ma j( T ) = 2 + 4 + 5 = 11. F or a cell □ in the Y oung diagram of λ we denote its ro w and column b y row( □ ) and col( □ ), resp ectiv ely , where the top-left cell is at row 1 and column 1. F or a ce ll □ ∈ λ its c ontent is defined as c ( □ ) = col( □ ) − row( □ ). The ho ok of □ consists of the cell □ itself, all cells to its righ t in the same row, and all cells b elo w it in the same column. The num b er of cells in the ho ok is the ho ok length , denoted h □ . SUPER MAJOR INDEX CYCLIC SIEVING 3 Example 2.2. Le t λ = (6 , 5 , 4 , 2 , 2 , 2). The conten ts and hook lengths of the cells are: 0 1 2 3 4 5 -1 0 1 2 3 -2 -1 0 1 -3 -2 -4 -3 -5 -4 11 10 6 5 3 1 9 8 4 3 1 7 6 2 1 4 3 3 2 2 1 The total num b er of standard Y oung tableaux of a given shap e is given by the celebrated ho ok-length formula of F rame, Robinson, and Thrall [FR T54]. Theorem 2.3 (Ho ok-Length F ormula) . L et λ b e a p artition of n . The numb er of standar d Y oung table aux of shap e λ is | SYT( λ ) | = n ! Q □ ∈ λ h □ , wher e the pr o duct is over al l c el ls u in the Y oung diagr am of λ . 2.2. Promotion. Sch¨ utzenberger promotion is a bijection on standard Y oung tableaux (SYT) of a fixed shape. Although we do not require the description of promotion itself for the statement of our main results, we recall its definition here for completeness, as it is the group action underlying our CSP . Giv en a tableau T ∈ SYT( λ ) of size n , its promotion pr( T ) is defined as follo ws: (1) Remo ve the en try 1, creating an empt y cell. (2) P erform jeu-de-taquin slides : at each step slide into the empty cell the smaller of the en tries immediately to its righ t or b elo w, un til the empt y cell reac hes an outer corner. (3) Insert n + 1 in to the empt y cell, yielding a tableau with en tries { 2 , 3 , . . . , n + 1 } . (4) Subtract 1 from every entry . The resulting tableau is the promoted tableau pr( T ). Promotion is in v ertible, preserv es shape, and acts with finite order; for rectangular shap es a × b it has order ab . See e.g. [Rho10, Theorem 1.1]. 2.3. q -analogues. W e recall standard q -analogues. F or a p ositiv e integer n , the q -inte ger is defined as [ n ] q : = 1 + q + · · · + q n − 1 = 1 − q n 1 − q , with [0] q = 0. The q -factorial is defined by [ n ] q ! : = Q n i =1 [ i ] q , with [0] q ! = 1. The q -binomial c o efficient is giv en b y  n k  q : = [ n ] q ! [ k ] q ![ n − k ] q ! for 0 ≤ k ≤ n , and 0 otherwise. The generating function for the ma jor index ov er SYT( λ ) is given b y a q -analogue of the ho ok-length form ula [Sta99, Corollary 7.21.5]. Theorem 2.4 ( q -Ho ok-Length F orm ula) . L et λ b e a p artition of n . The major index gener ating function is given by (1) f λ ( q ) = q κ ( λ ) [ n ] q ! Q □ ∈ λ [ h □ ] q , wher e κ ( λ ) = P i ≥ 1 ( i − 1) λ i . 4 STEPHAN PF ANNERER 2.4. The Cyclic Sieving Phenomenon. Let X b e a finite set of combinatorial ob jects, let C = ⟨ g ⟩ b e a cyclic group of order n acting on X , and let P ( q ) ∈ N [ q , q − 1 ] b e a Laurent p olynomial 2 with non-negative integer co efficien ts. Let ξ b e a complex primitiv e n -th ro ot of unit y . Definition 2.5 ([RSW04]) . The triple ( X , C , P ) is said to exhibit the cyclic sieving phenom- enon (CSP) if for all in tegers d ≥ 0, |{ x ∈ X : g d · x = x }| = P ( ξ d ) . W e also write X g for the set of fixed p oin ts of g on X and call the triple ( X , C, P ) a CSP triple if it exhibits the cyclic sieving phenomenon. Recall t w o fundamental CSPs for subsets and rectangular standard Y oung tableaux that w e will utilize. Theorem 2.6 ([RSW04]) . L et 0 ≤ k ≤ n . The triple  [ n ] k  , ⟨ cyc ⟩ ,  n k  q ! exhibits the cyclic sieving phenomenon, wher e cyc acts on k -element subsets of [ n ] by cyclic shift, i.e., de cr ementing e ach entry gr e ater than 1 by 1 and changing 1 to n . In p articular, for a divisor d | n and a primitive n -th r o ot of unity ξ , let s = n/d . Then (2)  n k  q = ξ d = (  d k/s  if s | k , 0 otherwise. Theorem 2.7 ([Rho10]) . L et λ ⊢ n b e a r e ctangular p artition. The triple  SYT( λ ) , ⟨ pr ⟩ , q − κ ( λ ) f λ ( q )  exhibits the cyclic sieving phenomenon, wher e pr acts by Sch¨ utzenb er ger pr omotion. Additionally , Rhoades’ result extends to the case of tuples of standard Y oung tableaux of same arbitrary shape λ , but here the group action is unkno wn. Theorem 2.8 ([APRU21]) . L et λ ⊢ n b e any p artition. Ther e exists a cyclic gr oup action of or der n on SYT( λ ) such that the triple (SYT( λ ) × SYT( λ ) × · · · × SYT( λ ) | {z } m c opies , C n , ( f λ ( q )) m ) exhibits the cyclic sieving phenomenon if and only if for al l d | n , ( f λ ( ξ d )) m is a non-ne gative inte ger. In p articular, if m is even, such a CSP always exists. W e can construct new CSP triples from existing ones via the Cartesian pro duct. Lemma 2.9 (Pro duct CSP) . L et ( X , C , P ) and ( Y , C, Q ) b e two triples exhibiting the cyclic sieving phenomenon with the same gr oup C = ⟨ g ⟩ . Then the triple ( X × Y , C, P · Q ) exhibits the CSP, wher e the gr oup action on X × Y is define d diagonal ly by g · ( x, y ) = ( g · x, g · y ) . Pr o of. Let d ≥ 0. The set of fixed p oin ts of g d on the pro duct is the Cartesian product of the fixed p oin t sets. That is ( X × Y ) g d = { ( x, y ) : g d x = x and g d y = y } = X g d × Y g d . T aking cardinalities, we ha ve | ( X × Y ) g d | = | X g d | · | Y g d | . 2 Note that classically , one considers polynomials and not Lauren t p olynomials. W e allo w the more general setup here, to av oid extra multiplications with appropriate pow ers of q n in our theorems. SUPER MAJOR INDEX CYCLIC SIEVING 5 Since the individual triples exhibit the CSP , | X g d | = P ( ξ d ) and | Y g d | = Q ( ξ d ). Therefore, | ( X × Y ) g d | = P ( ξ d ) Q ( ξ d ) = ( P · Q )( ξ d ) . Th us, the pro duct p olynomial correctly enumerates the fixed p oin ts. □ 2.5. Border Strip T ableaux. The ev aluation of the standard q -hook length form ula at ro ots of unity is closely related to the enumeration of border strip tableaux. Definition 2.10. A b or der strip is a connected sk ew shap e that does not contain a 2 × 2 square. A b or der strip table au of shape λ and t yp e µ = ( µ 1 , . . . , µ k ) is a filling of the Y oung diagram of λ with the num b ers 1 , . . . , k such that (1) the entries are w eakly increasing in eac h row from left to right, and (2) the entries are w eakly increasing in eac h column from top to b ottom, and (3) the cells filled with i form a b order strip of size µ i . W e denote b y BST( λ, k ) the set of b order strip tableaux of shap e λ such that all b order strips ha ve size k . Example 2.11. A b order strip tableau of shap e (6 , 5 , 4 , 2 , 2 , 2) with strips of length 3 is: 7 7 7 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 Theorem 2.12 ([JK84, Theorem 2.7.27].) . L et λ ⊢ n and let ξ b e a primitive d -th r o ot of unity. If d do es not divide n , then f λ ( ξ ) = 0 . If d | n , then (3) f λ ( ξ ) = ϵ λ | BST( λ, d ) | , wher e ϵ λ ∈ {− 1 , 1 } is a sign dep ending on the shap e λ . The crucial step in our pro of of the main theorems relies on the behavior of cell conten ts for shap es λ with BST( λ, m )  = ∅ . Lemma 2.13. L et λ b e a p artition of n . L et s b e a divisor of n such that BST( λ, s )  = ∅ . Then the multiset of c ontents { { c ( □ ) (mo d s ) : □ ∈ λ } } c onsists of the values { 0 , 1 , . . . , s − 1 } , e ach app e aring with multiplicity n/s . Pr o of. This follo ws from the structural prop erties of b order strip decomp ositions. If a shape can be tiled by b order strips of length s , the con tents of the cells in an y single border strip are distinct mo dulo s and co ver the full set of residues { 0 , . . . , s − 1 } exactly once. Since there are n/s strips in the decomp osition, the total collection of con tents co vers each residue n/s times. □ 3. Signed St andard T ableaux and Super Major Index Let λ ⊢ n b e a partition. A signe d standar d table au of shap e λ is a pair T = ( T + , D ), where: • T + ∈ SYT( λ ) is a standard Y oung tableau of shap e λ , and • D ⊆ { 1 , 2 , . . . , n } is a subset of entries, which we refer to as the ne gative entries of T , denoted by NEG( T ). The set of all signed standard tableaux of shap e λ is denoted by SYT ± ( λ ). The subset of tableaux with exactly k negative entries is denoted b y SYT ± k ( λ ). W e define a cyclic action pr on SYT ± ( λ ) by acting on the comp onents: pr( T ) : = (pr( T + ) , cyc( D )) . Definition 3.1 ([AS25]) . An integer i ∈ { 1 , . . . , n − 1 } is a sup er desc ent of T ∈ SYT ± ( λ ) if either: (1) i ∈ DES( T + ) and i + 1 / ∈ NEG( T ), or (2) i / ∈ DES( T + ) and i ∈ NEG( T ). 6 STEPHAN PF ANNERER The sup er major index of T , denoted ma j( T ), is the sum of its super descen ts. W e denote with f λ ± k ( q ) = X T ∈ SYT ± k ( λ ) q ma j( T ) the generating function for the sup er ma jor index o ver signed tableaux with k negative entries and with f λ ( q , t ) : = X k f λ ± k ( q ) t k = X T q ma j( T ) t | NEG( T ) | the biv ariate generating function summing o ver all signed standard tableaux that also trac ks the n umber of negative en tries with t . Example 3.2. C onsider the tableau from the previous Example 2.1. T + = 1 2 4 3 5 6 7 , NEG( T ) = { 3 , 6 } . W e ma y visualize this b y marking negative entries (e.g., with an ov erline): T = 1 2 4 3 5 6 7 The descents of T + are DES( T + ) = { 2 , 4 , 5 } . Let us compute the super descen ts: • i = 1: 1 / ∈ DES, 1 / ∈ NEG. No. • i = 2: 2 ∈ DES, 3 ∈ NEG. No. • i = 3: 3 / ∈ DES, 3 ∈ NEG. (Condition 2 holds). Y es . • i = 4: 4 ∈ DES, 5 / ∈ NEG. (Condition 1 holds). Y es . • i = 5: 5 ∈ DES, 6 ∈ NEG. No. • i = 6: 6 / ∈ DES, 6 ∈ NEG. (Condition 2 holds). Y es . Th us, the sup er descent set is { 3 , 4 , 6 } and ma j( T ) = 13. Before analyzing the super ma jor index, w e observe that Lemma 2.9 immediately provides a cyclic sieving phenomenon for super tableaux, viewing them simply as the Cartesian product SYT( λ ) ×  [ n ] k  . Prop osition 3.3 (T rivial Super CSP) . L et λ ⊢ n b e a r e ctangular p artition. The triple SYT ± k ( λ ) , ⟨ pr × cyc ⟩ , q − κ ( λ ) f λ ( q ) ·  n k  q ! exhibits the cyclic sieving phenomenon. R emark 3.4 . While v alid, this p olynomial is distinct from the one in Theorem 1.1. F or instance consider the partition λ = (3 , 3) and k = 1. The generating function for the classical ma jor index is f (3 , 3) ( q ) = q 3 + q 5 + q 6 + q 7 + q 9 . The p olynomial for the “T rivial Super CSP” given by Prop osition 3.3 is: P trivial ( q ) = q − 3 f (3 , 3) ( q )[6] q = 1 + q + 2 q 2 + 3 q 3 + 4 q 4 + 4 q 5 + 4 q 6 + 4 q 7 + 3 q 8 + 2 q 9 + q 10 + q 11 . F or this example the super ma jor index generating function is: f (3 , 3) ± 1 ( q ) = q 2 + 2 q 3 + 3 q 4 + 4 q 5 + 5 q 6 + 5 q 7 + 4 q 8 + 3 q 9 + 2 q 10 + q 11 . Multiplying by the t wist q γ (6 , 1) − κ (3 , 3) = q − 3 yields our main sieving polynomial: P super ( q ) = q − 1 + 2 + 3 q + 4 q 2 + 5 q 3 + 5 q 4 + 4 q 5 + 3 q 6 + 2 q 7 + q 8 . SUPER MAJOR INDEX CYCLIC SIEVING 7 Armon and Sw anson prov ed that the sup er ma jor index generating function has a pro duct form ula that refines the q -hook length formula b y incorporating the con tents of the cells. Theorem 3.5 ([AS25, Theorem 1.6]) . L et λ b e a p artition of n . The sup er major index gener- ating function is given by (4) f λ ( q , t ) = [ n ] q ! Y □ ∈ λ q row( □ ) − 1 + tq col( □ ) − 1 [ h □ ] q . F rom this we obtain that the generating function f λ ( q , t ) : = P T q ma j( T ) t | NEG( T ) | factors in a wa y crucial for our analysis. Corollary 3.6. The sup er major index gener ating function factors as: (5) f λ ( q , t ) = f λ ( q ) Y □ ∈ λ (1 + tq c ( □ ) ) , wher e c ( □ ) is the c ontent of the c el l □ . Pr o of. W e factor out q row( □ ) − 1 from the n umerator of eac h term in the product in (4): q row( □ ) − 1 + tq col( □ ) − 1 = q row( □ ) − 1 (1 + tq col( □ ) − row( □ ) ) = q row( □ ) − 1 (1 + tq c ( □ ) ) , where c ( □ ) = col( □ ) − row( □ ) is the conten t of the cell □ . Substituting this bac k into the expression for f λ ( q , t ), w e obtain: f λ ( q , t ) = [ n ] q ! Y □ ∈ λ q row( □ ) − 1 (1 + tq c ( □ ) ) [ h □ ] q = [ n ] q ! Q □ ∈ λ q row( □ ) − 1 Q □ ∈ λ [ h □ ] q ! Y □ ∈ λ (1 + tq c ( □ ) ) . The first factor can be simplified by observing that P □ ∈ λ (ro w ( □ ) − 1) = P i ≥ 1 λ i ( i − 1) = κ ( λ ). Th us, [ n ] q ! q P □ ∈ λ (row( □ ) − 1) Q □ ∈ λ [ h □ ] q = q κ ( λ ) [ n ] q ! Q □ ∈ λ [ h □ ] q = f λ ( q ) , b y the q -Hook-Length F ormula (1). □ 4. Proof of the Main Theorems In this section, w e first unify Theorems 1.1 and 1.2 and then pro ceed to pro v e the unified statemen t. The key idea is to leverage the pro duct formula for the sup er ma jor index generating function and analyze its ev aluation at ro ots of unit y in relation to border strip tableaux. Theorem 4.1. L et λ b e a p artition of n , m ≥ 1 , and α ∈ Z , such ther e exists a cyclic gr oup action of or der n gener ate d by g on SYT( λ ) m such that the triple (SYT( λ ) m , ⟨ g ⟩ , q α ( f λ ( q )) m ) exhibits the cyclic sieving phenomenon. Then the triple (SYT ± k ( λ ) m , ⟨ g × cyc m ⟩ , q α ( q γ ( n,k ) f λ ± k ( q )) m ) also exhibits the cyclic sieving phenomenon. This theorem serves as a general lifting theorem from which our main theorems follow as sp ecial cases. Pr o of of The or ems 1.1 and 1.2. Rectangular Case. F or a rectangular partition λ , Theorem 2.7 pro vides a CSP for (SYT( λ ) , ⟨ pr ⟩ , q − κ ( λ ) f λ ( q )). Applying Theorem 4.1 with m = 1, g = pr, and α = − κ ( λ ) yields the CSP for signed tableaux as stated in Theorem 1.1. Non-Rectangular Case. F or an arbitrary partition λ , Theorem 2.8 guarantees a CSP for (SYT( λ ) m , C n , ( f λ ( q )) m ) whenever the ev aluations are non-negativ e. Applying Theorem 4.1 with this base case (setting α = 0) directly implies Theorem 1.2. □ 8 STEPHAN PF ANNERER Before proving Theorem 4.1, w e start by ev aluating the pro duct term in (5) at a ro ot of unity . Prop osition 4.2. L et ζ b e a primitive s -th r o ot of unity. Then (6) s − 1 Y j =0 (1 + tζ j ) = 1 − ( − t ) s . Pr o of. W e start with the factorization of x s − 1 = Q s − 1 j =0 ( x − ζ j ). Substituting x = − 1 /t giv es ( − 1 /t ) s − 1 = s − 1 Y j =0 ( − 1 /t − ζ j ) = ( − 1 /t ) s s − 1 Y j =0 (1 + tζ j ) . Multiplying by ( − t ) s yields the desired identit y . □ Prop osition 4.3. L et λ ⊢ n b e a p artition. L et d | n and let ξ b e a primitive n -th r o ot of unity. Define s = n/d . If BST( λ, s )  = ∅ , then Y □ ∈ λ (1 + t ( ξ d ) c ( □ ) ) = (1 − ( − t ) s ) d . Pr o of. Let ζ = ξ d . Note that ζ is a primitive s -th ro ot of unit y . As BST( λ, s )  = ∅ , the exp onen ts c ( □ ) modulo s are equidistributed, b y Lemma 2.13. Thus, the pro duct factors in to groups based on the residues modulo s : Y □ ∈ λ (1 + tζ c ( □ ) ) =   s − 1 Y j =0 (1 + tζ j )   d . By (6), this simplifies to (1 − ( − t ) s ) d . □ W e can now pro ve the unified theorem: Pr o of of The or em 4.1. W e m ust show that for an y d ≥ 0, the ev aluation of the p olynomial at q = ξ d equals the num b er of fixed p oints of g d × (cyc m ) d on SYT ± k ( λ ) m . Since b oth the n umber of fixed points of a cyclic group elemen t g d and the ev aluation of the p olynomial at ξ d only dep end on gcd( n, d ), it is sufficien t to prov e the identit y for all d that divide n . Let d be a divisor of n , and let s = n/d . The root of unity is ζ = ξ d , a primitiv e s -th root of unit y . Let Fix( d ) denote the num b er of fixed p oints of the action of g d × (cyc m ) d on SYT ± k ( λ ) m . This is the pro duct of the fixed p oin t counts on the comp onen ts. By assumption, the num b er of fixed points of g d on SYT( λ ) m is given by    (SYT( λ ) m ) g d    = ζ α ( f λ ( ζ )) m . F or the action of (cyc m ) d on (  [ n ] k  ) m , the num b er of fixed p oin ts is  n k  q = ζ ! m b y Theorem 2.6 and lemma 2.9. Using the ev aluation of the q -binomial co efficients from (2) w e hav e Fix( d ) = ( ζ α  f λ ( ζ )  d k/s   m if s | k 0 otherwise. No w w e ev aluate the p olynomial P ( q ) = q α ( q γ ( n,k ) f λ ± k ( q )) m at q = ζ . Using the factorization (5), we hav e P ( ζ ) = ζ α ζ γ ( n,k ) f λ ( ζ )[ t k ] Y □ ∈ λ (1 + tζ c ( □ ) ) ! m . If BST( λ, s ) = ∅ , then by (3), f λ ( ζ ) = 0 and th us, Fix( d ) = P ( ζ ) = 0. SUPER MAJOR INDEX CYCLIC SIEVING 9 If BST( λ, s )  = ∅ , then by Prop osition 4.3, [ t k ] Y □ ∈ λ (1 + tζ c ( □ ) ) = [ t k ](1 − ( − t ) s ) d = (  d k/s  ( − 1) k/s + k if s | k 0 otherwise. Then, in the case where k is not divisible by s w e directly get P ( ζ ) = Fix( d ) = 0. In the other case, let j = k /s . It remains to chec k that the twist factor ζ γ ( n,k ) accoun ts for the sign difference. Indeed, using ξ n = 1 and ζ = ξ d , we ha ve ζ γ ( n,k ) = ξ d ( n − 1) ( k 2 ) = ζ − ( k 2 ) . W e c heck that ζ − ( k 2 ) = ( − 1) j + k . If s is o dd, then s |  k 2  and j + k is even, so b oth sides are 1. If s is even, then k is ev en and ζ s/ 2 = − 1. So ζ − ( k 2 ) = ( − 1) − j ( k − 1) = ( − 1) j = ( − 1) j + k . Therefore, P ( ζ ) = Fix( d ). □ A cknowledgements The author was partially supp orted b y Oliver P echenik’s Disco very Grant (RGPIN-2021- 02391) and Launch Supplement (DGECR-2021-00010) from the Natural Sciences and Engi- neering Research Council of Canada and was also partially supp orted by Oly a Mandelshtam’s Disco very Grant (R GPIN-2021-02568). W e are grateful for helpful conv ersations with Josh Swanson and Oliver Pec henik. References [APRU21] P . Alexandersson, S. Pfannerer, M. Rubey , and J. Uhlin. Skew characters and cyclic sieving. F orum of Mathematics, Sigma , 9:e41, 2021. [AS25] S. Armon and J. Swanson. Super ma jor index and Thrall’s problem. Algebr aic Combinatorics , 8(3):795–815, June 2025. [FR T54] S. F rame, G. Robinson, and R. Thrall. The ho ok graphs of the symmetric group. Canadian Journal of Mathematics , 6:316–324, 1954. [JK84] G. James and A. Kerb er. The R epresentation The ory of the Symmetric Gr oup . Cambridge Universit y Press, December 1984. 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